ACI 351.3R-04
Foundations for Dynamic Equipment
Reported by ACI Committee 351
James P. Lee*
Chair
Yelena S. Golod*
Secretary
William L. Bounds*
Fred G. Louis
Abdul Hai Sheikh
William D. Brant
Jack Moll
Anthony J. Smalley
Shu-jin Fang
Ira W. Pearce
*
Shraddhakar Harsh
Andrew Rossi
Charles S. Hughes
Robert L. Rowan, Jr.‡
Erick Larson
William E. Rushing, Jr.
*
Members of the editorial subcommittee.
†
Chair of subcommittee that prepared this
‡Past chair.
Philip A. Smith
W. Tod Sutton†
F. Alan Wiley
report.
This report presents to industry practitioners the various design criteria
and methods and procedures of analysis, design, and construction applied
to dynamic equipment foundations.
Keywords: amplitude; concrete; foundation; reinforcement; vibration.
CONTENTS
Chapter 1—Introduction, p. 351.3R-2
1.1—Background
1.2—Purpose
1.3—Scope
1.4—Notation
Chapter 2—Foundation and machine types,
p. 351.3R-4
2.1—General considerations
ACI Committee Reports, Guides, Standard Practices, and
Commentaries are intended for guidance in planning,
designing, executing, and inspecting construction. This
document is intended for the use of individuals who are
competent to evaluate the significance and limitations of its
content and recommendations and who will accept
responsibility for the application of the material it contains.
The American Concrete Institute disclaims any and all
responsibility for the stated principles. The Institute shall not
be liable for any loss or damage arising therefrom.
Reference to this document shall not be made in contract
documents. If items found in this document are desired by the
Architect/Engineer to be a part of the contract documents, they
shall be restated in mandatory language for incorporation by
the Architect/Engineer.
It is the responsibility of the user of this document to
establish health and safety practices appropriate to the specific
circumstances involved with its use. ACI does not make any
representations with regard to health and safety issues and the
use of this document. The user must determine the
applicability of all regulatory limitations before applying the
document and must comply with all applicable laws and
regulations, including but not limited to, United States
Occupational Safety and Health Administration (OSHA)
health and safety standards.
2.2—Machine types
2.3—Foundation types
Chapter 3—Design criteria, p. 351.3R-7
3.1—Overview of design criteria
3.2—Foundation and equipment loads
3.3—Dynamic soil properties
3.4—Vibration performance criteria
3.5—Concrete performance criteria
3.6—Performance criteria for machine-mounting systems
3.7—Method for estimating inertia forces from multicylinder machines
Chapter 4—Design methods and materials,
p. 351.3R-26
4.1—Overview of design methods
4.2—Impedance provided by the supporting media
4.3—Vibration analysis
4.4—Structural foundation design and materials
4.5—Use of isolation systems
4.6—Repairing and upgrading foundations
4.7—Sample impedance calculations
Chapter 5—Construction considerations,
p. 351.3R-53
5.1—Subsurface preparation and improvement
5.2—Foundation placement tolerances
5.3—Forms and shores
5.4—Sequence of construction and construction joints
5.5—Equipment installation and setting
5.6—Grouting
5.7—Concrete materials
5.8—Quality control
ACI 351.3R-04 became effective May 3, 2004.
Copyright © 2004, American Concrete Institute.
All rights reserved including rights of reproduction and use in any form or by any
means, including the making of copies by any photo process, or by electronic or
mechanical device, printed, written, or oral, or recording for sound or visual reproduction or for use in any knowledge or retrieval system or device, unless permission in
writing is obtained from the copyright proprietors.
351.3R-1
351.3R-2
ACI COMMITTEE REPORT
Chapter 6—References, p. 351.3R-57
6.1—Referenced standards and reports
6.2—Cited references
6.3—Software sources and other references
6.4—Terminology
CHAPTER 1—INTRODUCTION
1.1—Background
Heavy machinery with reciprocating, impacting, or rotating
masses requires a support system that can resist dynamic
forces and the resulting vibrations. When excessive, such
vibrations may be detrimental to the machinery, its support
system, and any operating personnel subjected to them.
Many engineers with varying backgrounds are engaged in
the analysis, design, construction, maintenance, and repair of
machine foundations. Therefore, it is important that the
owner/operator, geotechnical engineer, structural engineer,
and equipment supplier collaborate during the design
process. Each of these participants has inputs and concerns
that are important and should be effectively communicated
with each other, especially considering that machine foundation
design procedures and criteria are not covered in building
codes and national standards. Some firms and individuals
have developed their own standards and specifications as a
result of research and development activities, field studies,
or many years of successful engineering or construction
practices. Unfortunately, most of these standards are not
available to many practitioners. As an engineering aid to
those persons engaged in the design of foundations for
machinery, the committee developed this document, which
presents many current practices for dynamic equipment
foundation engineering and construction.
1.2—Purpose
The committee presents various design criteria and
methods and procedures of analysis, design, and construction
currently applied to dynamic equipment foundations by
industry practitioners.
This document provides general guidance with reference
materials, rather than specifying requirements for adequate
design. Where the document mentions multiple design
methods and criteria in use, factors, which may influence the
choice, are presented.
1.3—Scope
This document is limited in scope to the engineering,
construction, repair, and upgrade of dynamic equipment
foundations. For the purposes of this document, dynamic
equipment includes the following:
1. Rotating machinery;
2. Reciprocating machinery; and
3. Impact or impulsive machinery.
1.4—Notation
[C]
=
[K]
=
[K*]
=
[k]
=
=
[kj′ ]
damping matrix
stiffness matrix
impedance with respect to CG
reduced stiffness matrix
battered pile stiffness matrix
[M]
[m]
[T]
[αir]
=
=
=
=
A
Ahead , Acrank
Ap
a, b
ao
Bc
Bi
Br
b1, b2
cgi
ci
ci*(adj)
=
=
=
=
=
=
=
=
=
=
=
=
cij
=
Di
Drod
d
dn
ds
Ep
em
ev
F
F1
Fblock
=
=
=
=
=
=
=
=
=
=
=
(Fbolt)CHG
=
(Fbolt)frame
=
FD
FGMAX
=
=
FIMAX
=
Fo
=
Fr
Fred
=
=
Frod
Fs
FTHROW
=
=
=
Funbalance
=
mass matrix
reduced mass matrix
transformation matrix for battered pile
matrix of interaction factors between any
two piles with diagonal terms αii = 1
displacement amplitude
head and crank areas, in.2 (mm2)
cross-sectional area of the pile
plan dimensions of a rectangular foundation
dimensionless frequency
cylinder bore diameter, in. (mm)
mass ratio for the i-th direction
ram weight, tons (kN)
0.425 and 0.687, Eq. (4.15d)
damping of pile group in the i-th direction
damping constant for the i-th direction
damping in the i-th direction adjusted for
material damping
equivalent viscous damping of pile j in the
i-th direction
damping ratio for the i-th direction
rod diameter, in. (mm)
pile diameter
nominal bolt diameter, in. (m)
displacement of the slide, in. (mm)
Young’s modulus of the pile
mass eccentricity, in. (mm)
void ratio
time varying force vector
correction factor
the force acting outwards on the block from
which concrete stresses should be calculated, lbf (N)
the force to be restrained by friction at the
cross head guide tie-down bolts, lbf (N)
the force to be restrained by friction at the
frame tie-down bolts, lbf (N)
damper force
maximum horizontal gas force on a throw
or cylinder, lbf (N)
maximum horizontal inertia force on a
throw or cylinder, lbf (N)
dynamic force amplitude (zero-to-peak),
lbf (N)
maximum horizontal dynamic force
a force reduction factor with suggested
value of 2, to account for the fraction of
individual cylinder load carried by the
compressor frame (“frame rigidity
factor”)
force acting on piston rod, lbf (N)
dynamic inertia force of slide, lbf (N)
horizontal force to be resisted by each
throw’s anchor bolts, lbf (N)
the maximum value from Eq. (3.18)
applied using parameters for a horizontal
compressor cylinder, lbf (N)
FOUNDATIONS FOR DYNAMIC EQUIPMENT
fi1, fi2
fm
fn
fo
G
Gave
Gc
GE
Gl
G pJ
Gs
Gz
H
Ii
Ip
i
i
K2
Keff
K*ij
Kn
Kuu
Kuψ
Kψψ
k
*
kei
kgi
ki
ki(adj)
ki*
ki*(adj)
kij
kj
kr
kst
kvj
L
LB
Li
l
lc
lp
Mh
= dimensionless stiffness and damping
functions for the i-th direction, piles
= frequency of motion, Hz
= system natural frequency (cycles per second)
= operating speed, rpm
= dynamic shear modulus of the soil
= the average value of shear modulus of the
soil over the pile length
= the average value of shear modulus of the
soil over the critical length
= pile group efficiency
= soil shear modulus at tip of pile
= torsional stiffness of the pile
= dynamic shear modulus of the embedment
(side) material
= the shear modulus at depth z = lc /4
= depth of soil layer
= mass moment of inertia of the machinefoundation system for the i-th direction
= moment of inertia of the pile cross section
= –1
= a directional indicator or modal indicator,
Eq. (4.48), as a subscript
= a parameter that depends on void ratio and
strain amplitude
= the effective bearing stiffness, lbf/in. (N/mm)
= impedance in the i-th direction with respect
to motion of the CG in j-th direction
= nut factor for bolt torque
= horizontal spring constant
= coupling spring constant
= rocking spring constant
= the dynamic stiffness provided by the
supporting media
= impedance in the i-th direction due to
embedment
= pile group stiffness in the i-th direction
= stiffness for the i-th direction
= stiffness in the i-th direction adjusted for
material damping
= complex impedance for the i-th direction
= impedance adjusted for material damping
= stiffness of pile j in the i-th direction
= battered pile stiffness matrix
= stiffness of individual pile considered in
isolation
= static stiffness constant
= vertical stiffness of a single pile
= length of connecting rod, in. (mm)
= the greater plan dimension of the foundation block, ft (m)
= length of the connecting rod of the crank
mechanism at the i-th cylinder
= depth of embedment (effective)
= critical length of a pile
= pile length
= hammer mass including any auxiliary
foundation, lbm (kg)
Mr
351.3R-3
= ram mass including dies and ancillary
parts, lbm (kg)
m
= mass of the machine-foundation system
md
= slide mass including the effects of any
balance mechanism, lbm (kg)
= rotating mass, lbm (kg)
mr
= reciprocating mass for the i-th cylinder
mrec,i
= rotating mass of the i-th cylinder
mrot,i
= effective mass of a spring
ms
= the number of bolts holding down one
(Nbolt)CHG
crosshead guide
(Nbolt)frame
= the number of bolts holding down the
frame, per cylinder
NT
= normal torque, ft-lbf (m-N)
Phead, Pcrank = instantaneous head and crank pressures,
psi (µPa)
= power being transmitted by the shaft at the
Ps
connection, horsepower (kilowatts)
R, Ri
= equivalent foundation radius
r
= length of crank, in. (mm)
= radius of the crank mechanism of the i-th
ri
cylinder
ro
= pile radius or equivalent radius
S
= press stroke, in. (mm)
= service factor, used to account for increasing
Sf
unbalance during the service life of the
machine, generally greater than or equal to 2
Si1, Si2
= dimensionless parameters (Table 4.2)
s
= distance between piles
T
= foundation thickness, ft (m)
= bolt torque, lbf-in. (N-m)
Tb
= minimum required anchor bolt tension
Tmin
t
= time, s
= the maximum allowable vibration, in. (mm)
Vmax
= shear wave velocity of the soil, ft/s (m/s)
Vs
v
= displacement amplitude
v′
= velocity, in./s (cm/s)
= post-impact hammer velocity, in./s (mm/s)
vh
= reference velocity = 18.4 ft/s (5.6 m/s)
vo
from a free fall of 5.25 ft (1.6 m)
= ram impact velocity, ft/s (m/s)
vr
W
= strain energy
= equipment weight at anchorage location
Wa
= weight of the foundation, tons (kN)
Wf
= bolt preload, lbf (N)
Wp
= rotating weight, lbf (N)
Wr
w
= soil weight density
X
= vector representation of time-dependent
displacements for MDOF systems
= distance along the crankshaft from the
Xi
reference origin to the i-th cylinder
x, z
= the pile coordinates indicated in Fig. 4.9
= pile location reference distances
xr, zr
= distance from the CG to the base support
yc
= distance from the CG to the level of
ye
embedment resistance
yp
= crank pin displacement in local Y-axis,
in. (mm)
351.3R-4
Zp
zp
α
α′
α1
αh
αi
α*ij
αuf
αuH
αv
αψH
αψM
β
βi
βj
βm
βp
δ
∆W
εir
ψi
γj
λ
µ
ν
νs
ρ
ρa
ρc
σo
ωi
ωm
ωn
ωo
ωsu, ωsv
ACI COMMITTEE REPORT
= piston displacement, in. (mm)
= crank pin displacement in local Z-axis, in.
(mm)
= the angle between a battered pile and
vertical
= modified pile group interaction factor
= coefficient dependent on Poisson’s ratio
as given in Table 4.1
= ram rebound velocity relative to impact
velocity
= the phase angle for the crank radius of the
i-th cylinder, rad
= complex pile group interaction factor for
the i-th pile to the j-th pile
= the horizontal interaction factor for fixedheaded piles (no head rotation)
= the horizontal interaction factor due to
horizontal force (rotation allowed)
= vertical interaction coefficient between
two piles
= the rotation due to horizontal force
= the rotation due to moment
= system damping ratio
= rectangular footing coefficients (Richart,
Hall, and Woods 1970), i = v, u, or ψ
= coefficient dependent on Poisson’s ratio
as given in Table 4.1, j = 1 to 4
= material damping ratio of the soil
= angle between the direction of the loading
and the line connecting the pile centers
= loss angle
= area enclosed by the hysteretic loop
= the elements of the inverted matrix [αir]–1
= reduced mode shape vector for the i-th
mode
= coefficient dependent on Poisson’s ratio
as given in Table 4.1, j = 1 to 4
= pile-soil stiffness ratio (Ep /Gl )
= coefficient of friction
= Poisson’s ratio of the soil
= Poisson’s ratio of the embedment (side)
material
= soil mass density (soil weight density/gravitational acceleration)
= Gave /Gl
= Gz /Gc
= probable confining pressure, lbf/ft2 (Pa)
= circular natural frequency for the i-th
mode
= circular frequency of motion
= circular natural frequencies of the system
= circular operating frequency of the
machine (rad/s)
= circular natural frequencies of a soil layer
in u and v directions
CHAPTER 2—FOUNDATION AND MACHINE TYPES
2.1—General considerations
The type, configuration, and installation of a foundation or
support structure for dynamic machinery may depend on the
following factors:
1. Site conditions such as soil characteristics, topography,
seismicity, climate, and other effects;
2. Machine base configuration such as frame size,
cylinder supports, pulsation bottles, drive mechanisms,
and exhaust ducts;
3. Process requirements such as elevation requirements
with respect to connected process equipment and hold-down
requirements for piping;
4. Anticipated loads such as the equipment static weight,
and loads developed during erection, startup, operation,
shutdown, and maintenance;
5. Erection requirements such as limitations or constraints
imposed by construction equipment, procedures, techniques,
or the sequence of erection;
6. Operational requirements such as accessibility, settlement limitations, temperature effects, and drainage;
7. Maintenance requirements such as temporary access,
laydown space, in-plant crane capabilities, and machine
removal considerations;
8. Regulatory factors or building code provisions such as
tied pile caps in seismic zones;
9. Economic factors such as capital cost, useful or anticipated life, and replacement or repair cost;
10. Environmental requirements such as secondary
containment or special concrete coating requirements; and
11. Recognition that certain machines, particularly large
reciprocating compressors, rely on the foundation to add
strength and stiffness that is not inherent in the structure of
the machine.
2.2—Machine types
2.2.1 Rotating machinery—This category includes gas
turbines, steam turbines, and other expanders; turbo-pumps
and compressors; fans; motors; and centrifuges. These
machines are characterized by the rotating motion of impellers or rotors.
Unbalanced forces in rotating machines are created when
the mass centroid of the rotating part does not coincide with
the center of rotation (Fig. 2.1). This dynamic force is a function
of the shaft mass, speed of rotation, and the magnitude of the
offset. The offset should be minor under manufactured
conditions when the machine is well balanced, clean, and
without wear or erosion. Changes in alignment, operation
near resonance, blade loss, and other malfunctions or
undesirable conditions can greatly increase the force applied
to its bearings by the rotor. Because rotating machines
normally trip and shut down at some vibration limit, a realistic continuous dynamic load on the foundation is that
resulting from vibration just below the trip level.
2.2.2 Reciprocating machinery—For reciprocating
machinery, such as compressors and diesel engines, a piston
moving in a cylinder interacts with a fluid through the
FOUNDATIONS FOR DYNAMIC EQUIPMENT
kinematics of a slider crank mechanism driven by, or
driving, a rotating crankshaft.
Individual inertia forces from each cylinder and each
throw are inherently unbalanced with dominant frequencies
at one and two times the rotational frequency (Fig. 2.2).
Reciprocating machines with more than one piston require
a particular crank arrangement to minimize unbalanced
forces and moments. A mechanical design that satisfies
operating requirements should govern. This leads to piston/
cylinder assemblies and crank arrangements that do not
completely counter-oppose; therefore, unbalanced loads
occur, which should be resisted by the foundation.
Individual cylinder fluid forces act outward on the
cylinder head and inward on the crankshaft (Fig. 2.2). For a
rigid cylinder and frame these forces internally balance, but
deformations of large machines can cause a significant
portion of the fluid load to be transmitted to the mounts and
into the foundation. Particularly on large reciprocating
compressors with horizontal cylinders, it is inappropriate
and unconservative to assume the compressor frame and
cylinder are sufficiently stiff to internally balance all forces.
Such an assumption has led to many inadequate mounts for
reciprocating machines.
2.2.3 Impulsive machinery—Equipment, such as forging
hammers and some metal-forming presses, operate with
regulated impacts or shocks between different parts of the
equipment. This shock loading is often transmitted to the
foundation system of the equipment and is a factor in the
design of the foundation.
Closed die forging hammers typically operate by dropping
a weight (ram) onto hot metal, forcing it into a predefined
shape. While the intent is to use this impact energy to form
and shape the material, there is significant energy transmission,
particularly late in the forming process. During these final
blows, the material being forged is cooling and less shaping
takes place. Thus, pre-impact kinetic energy of the ram
converts to post-impact kinetic energy of the entire forging
hammer. As the entire hammer moves downward, it
becomes a simple dynamic mass oscillating on its supporting
medium. This system should be well damped so that the
oscillations decay sufficiently before the next blow. Timing
of the blows commonly range from 40 to 100 blows per min.
The ram weights vary from a few hundred pounds to 35,000 lb
(156 kN). Impact velocities in the range of 25 ft/s (7.6 m/s)
are common. Open die hammers operate in a similar fashion
but are often of two-piece construction with a separate
hammer frame and anvil.
Forging presses perform a similar manufacturing function
as forging hammers but are commonly mechanically or
hydraulically driven. These presses form the material at low
velocities but with greater forces. The mechanical drive
system generates horizontal dynamic forces that the engineer
should consider in the design of the support system. Rocking
stability of this construction is important. Figure 2.3 shows a
typical horizontal forcing function through one full stroke of
a forging press.
Mechanical metal forming presses operate by squeezing
and shearing metal between two dies. Because this equip-
351.3R-5
Fig. 2.1—Rotating machine diagram.
Fig. 2.2—Reciprocating machine diagram.
Fig. 2.3—Forcing function for a forging press.
ment can vary greatly in size, weight, speed, and operation,
the engineer should consider the appropriate type. Speeds
can vary from 30 to 1800 strokes per min. Dynamic forces from
the press develop from two sources: the mechanical balance of
the moving parts in the equipment and the response of the
press frame as the material is sheared (snap-through forces).
Imbalances in the mechanics of the equipment can occur both
horizontally and vertically. Generally high-speed equipment
is well balanced. Low-speed equipment is often not balanced
because the inertia forces at low speeds are small. The
dynamic forces generated by all of these presses can be
significant as they are transmitted into the foundation and
propagated from there.
2.2.4 Other machine types—Other machinery generating
dynamic loads include rock crushers and metal shredders.
While part of the dynamic load from these types of equipment
tend to be based on rotating imbalances, there is also a
351.3R-6
ACI COMMITTEE REPORT
Fig. 2.4—Block-type foundation.
Fig. 2.8—Spring-mounted block formation.
Fig. 2.5—Combined block foundation.
Fig. 2.6—Tabletop foundation.
Fig. 2.7—Tabletop with isolators.
random character to the dynamic signal that varies with the
particular operation.
2.3—Foundation types
2.3.1 Block-type foundation (Fig. 2.4)—Dynamic machines
are preferably located close to grade to minimize the elevation
difference between the machine dynamic forces and the center
of gravity of the machine-foundation system. The ability to use
such a foundation primarily depends on the quality of near
surface soils. Block foundations are nearly always designed as
rigid structures. The dynamic response of a rigid block
foundation depends only on the dynamic load, foundation’s
mass, dimensions, and soil characteristics.
2.3.2 Combined block-type foundation (Fig. 2.5)—
Combined blocks are used to support closely spaced
machines. Combined blocks are more difficult to design
because of the combination of forces from two or more
machines and because of a possible lack of stiffness of a
larger foundation mat.
2.3.3 Tabletop-type foundation (Fig. 2.6)—Elevated
support is common for large turbine-driven equipment such
as electric generators. Elevation allows for ducts, piping, and
ancillary items to be located below the equipment. Tabletop
structures are considered to be flexible, hence their response
to dynamic loads can be quite complex and depend both on
the motion of its discreet elements (columns, beams, and
footing) and the soil upon which it is supported.
2.3.4 Tabletop with isolators (Fig. 2.7)—Isolators (springs
and dampers) located at the top of supporting columns are
sometimes used to minimize the response to dynamic loading.
The effectiveness of isolators depends on the machine speed
and the natural frequency of the foundation. Details of this
type of support are provided in Section 4.5.
2.3.5 Spring-mounted equipment (Fig. 2.8)—Occasionally
pumps are mounted on springs to minimize thermal forces
from connecting piping. The springs are then supported on a
block-type foundation. This arrangement has a dynamic
effect similar to that for tabletops with vibration isolators.
Other types of equipment are spring mounted to limit the
transmission of dynamic forces.
2.3.6 Inertia block in structure (Fig. 2.9)—Dynamic equipment on a structure may be relatively small in comparison to the
overall size of the structure. In this situation, dynamic machines
are usually designed with a supporting inertia block to alter
natural frequencies away from machine operating speeds and
resist amplitudes by increasing the resisting inertia force.
2.3.7 Pile foundations (Fig. 2.10)—Any of the previously
mentioned foundation types may be supported directly on soil
or on piles. Piles are generally used where soft ground condi-
FOUNDATIONS FOR DYNAMIC EQUIPMENT
Fig. 2.9—Inertia block in structure.
Fig. 2.10—Pile-supported foundation.
tions result in low allowable contact pressures and excessive
settlement for a mat-type foundation. Piles use end bearing,
frictional side adhesion, or a combination of both to transfer
axial loads into the underlying soil. Transverse loads are
resisted by soil pressure bearing against the side of the pile
cap or against the side of the piles. Various types of piles are
used including drilled piers, auger cast piles, and driven piles.
CHAPTER 3—DESIGN CRITERIA
3.1—Overview of design criteria
The main issues in the design of concrete foundations that
support machinery are defining the anticipated loads, establishing the performance criteria, and providing for these
through proper proportioning and detailing of structural
members. Yet, behind this straightforward definition lies the
need for careful attention to the interfaces between machine,
mounting system, and concrete foundation.
The loads on machine foundations may be both static and
dynamic. Static loads are principally a function of the
weights of the machine and all its auxiliary equipment.
Dynamic loads, which occur during the operation of the
machine, result from forces generated by unbalance, inertia
of moving parts, or both, and by the flow of fluid and gases
for some machines. The magnitude of these dynamic loads
primarily depends upon the machine’s operating speed and
the type, size, weight, and arrangement (position) of moving
parts within the casing.
The basic goal in the design of a machine foundation is to
limit its motion to amplitudes that neither endanger the satisfactory operation of the machine nor disturb people working in
the immediate vicinity (Gazetas 1983). Allowable amplitudes
depend on the speed, location, and criticality or function of
351.3R-7
the machine. Other limiting dynamic criteria affecting the
design may include avoiding resonance and excessive transmissibility to the supporting soil or structure. Thus, a key
ingredient to a successful design is the careful engineering
analysis of the soil-foundation response to dynamic loads
from the machine operation.
The foundation’s response to dynamic loads can be significantly influenced by the soil on which it is constructed.
Consequently, critical soil parameters, such as the dynamic
soil shear modulus, are preferably determined from a field
investigation and laboratory tests rather than relying on
generalized correlations based on broad soil classifications.
Due to the inherent variability of soil, the dynamic response
of machine foundations is often evaluated using a range of
values for the critical soil properties.
Furthermore, a machinery support structure or foundation is
designed with adequate structural strength to resist the worst
possible combination of loads occurring over its service life.
This often includes limiting soil-bearing pressures to well
within allowable limits to ensure a more predictable dynamic
response and prevent excessive settlements and soil failures.
Additionally, concrete members are designed and detailed to
prevent cracking due to fatigue and stress reversals caused by
dynamic loads, and the machine’s mounting system is designed
and detailed to transmit loads from the machine into the
foundation, according to the criteria in Section 3.6.
3.2—Foundation and equipment loads
Foundations supporting reciprocating or rotating compressors,
turbines, generators and motors, presses, and other machinery
should withstand all the forces that may be imposed on them
during their service life. Machine foundations are unique
because they may be subjected to significant dynamic loads
during operation in addition to normal design loads of
gravity, wind, and earthquake. The magnitude and characteristics of the operating loads depend on the type, size,
speed, and layout of the machine.
Generally, the weight of the machine, center of gravity,
surface areas, and operating speeds are readily available
from the manufacturer of the machine. Establishing appropriate values for dynamic loads is best accomplished through
careful communication and clear understanding between the
machine manufacturer and foundation design engineer as to
the purpose, and planned use for the requested information,
and the definition of the information provided. It is in the best
interests of all parties (machine manufacturer, foundation
design engineer, installer, and operator) to ensure effective
definition and communication of data and its appropriate use.
Machines always experience some level of unbalance, vibration, and force transmitted through the bearings. Under some
off-design conditions, such as wear, the forces may increase
significantly. The machine manufacturer and foundation
design engineer should work together so that their combined
knowledge achieves an integrated system structure which
robustly serves the needs of its owner and operator and withstands all expected loads.
Sections 3.2.1 to 3.2.6 provide commonly used methods
for determining machine-induced forces and other design
351.3R-8
ACI COMMITTEE REPORT
loads for foundations supporting machinery. They include
definitions and other information on dynamic loads to be
requested from the machine manufacturer and alternative
assumptions to apply when such data are unavailable or are
under-predicted.
3.2.1 Static loads
3.2.1.1 Dead loads—A major function of the foundation
is to support gravity (dead) loads due to the weight of the
machine, auxiliary equipment, pipe, valves, and deadweight
of the foundation structure. The weights of the machine components are normally supplied by the machine manufacturer. The
distribution of the weight of the machine on the foundation
depends on the location of support points (chocks, soleplates)
and on the flexibility of the machine frame. Typically, there
are multiple support points, and, thus, the distribution is
statically indeterminate. In many cases, the machine manufacturer provides a loading diagram showing the vertical loads at
each support point. When this information is not available, it is
common to assume the machine frame is rigid and that its
weight is appropriately distributed between support points.
3.2.1.2 Live loads—Live loads are produced by personnel,
tools, and maintenance equipment and materials. The live loads
used in design should be the maximum loads expected during
the service life of the machine. For most designs, live loads are
uniformly distributed over the floor areas of platforms of
elevated support structures or to the access areas around atgrade foundations. Typical live loads vary from 60 lbf/ft2
(2.9 kPa) for personnel to as much as 150 lbf/ft2 (7.2 kPa) for
maintenance equipment and materials.
3.2.1.3 Wind loads—Loads due to wind on the surface
areas of the machine, auxiliary equipment, and the support
foundation are based on the design wind speed for the particular site and are normally calculated in accordance with the
governing local code or standard. Wind loads rarely govern
the design of machine foundations except, perhaps, when the
machine is located in an enclosure that is also supported by
the foundation.
When designing machine foundations and support structures,
most practitioners use the wind load provisions of ASCE 7. The
analytical procedure of ASCE 7 provides wind pressures and
forces for use in the design of the main wind-force resisting
systems and anchorage of machine components.
Most structural systems involving machines and machine
foundations are relatively stiff (natural frequency in the
lateral direction greater than 1 Hz). Consequently, the
systems can be treated as rigid with respect to the wind gust
effect factor, and simplified procedures can be used. If the
machine is supported on flexible isolators and is exposed to
the wind, the rigid assumption may not be reasonable, and
more elaborate treatment of the gust effects is necessary as
described in ASCE 7 for flexible structural systems.
Appropriate consideration of the exposure conditions and
importance factors is also required to be consistent with the
facilities requirements.
3.2.1.4 Seismic loads—Machinery foundations located
in seismically active regions are analyzed for seismic loads.
Before 2000, these loads were determined in accordance
with methods prescribed in one of various regional building
codes (such as the UBC, the SBC, or the NBC) and standards
such as ASCE 7 and SEAOC Blue Book.
The publication of the IBC 2000 provides building officials
with the opportunity to replace the former regional codes
with a code that has nationwide applicability. The seismic
requirements in IBC 2000 and ASCE 7-98 are essentially
identical, as both are based on the 1997 NEHRP (FEMA 302)
provisions.
The IBC and its reference documents contain provisions
for design of nonstructural components, including dynamic
machinery, for seismic loads. For machinery supported
above grade or on more flexible elevated pedestals, seismic
amplification factors are also specified.
3.2.1.5 Static operating loads—Static operating loads
include the weight of gas or liquid in the machinery equipment
during normal operation and forces, such as the drive torque
developed by some machines at the connection between the
drive mechanism and driven machinery. Static operating
loads can also include forces caused by thermal growth of
the machinery equipment and connecting piping. Timevarying (dynamic) loads generated by machines during
operation are covered elsewhere in this report.
Machines such as compressors and generators require
some form of drive mechanism, either integral with the
machine or separate from it. When the drive mechanism is
nonintegral, such as a separate electric motor, reciprocating
engine, and gas or steam turbine, it produces a net external
drive torque on the driven machine. The torque is equal in
magnitude and opposite in direction on the driver and driven
machine. The normal torque (sometimes called drive torque)
is generally applied to the foundation as a static force couple
in the vertical direction acting about the centerline of the
shaft of the machine. The magnitude of the normal torque is
often computed from the following formula
( 5250 ) ( P s )
NT = -------------------------- lbf-ft
fo
(3-1)
( 9550 ) ( P s )
NT = -------------------------- N-m
fo
where
NT =
Ps
=
normal torque, ft-lbf (m-N);
power being transmitted by the shaft at the
connection, horsepower (kilowatts); and
fo
= operating speed, rpm.
The torque load is generally resolved into a vertical force
couple by dividing it by the center-to-center distance between
longitudinal soleplates or anchor points (Fig. 3.1(a)). When the
machine is supported by transverse soleplates only, the
torque is applied along the width of the soleplate assuming a
straight line variation of force (Fig. 3.1(b)). Normal torque
can also be caused by jet forces on turbine blades. In this case
it is applied to the foundation in the opposite direction from
the rotation of the rotor.
The torque on a generator stator is applied in the same
direction as the rotation of the rotor and can be high due to
FOUNDATIONS FOR DYNAMIC EQUIPMENT
startup or an electrical short circuit. Startup torque, a property
of electric motors, should be obtained from the motor
manufacturer. The torque created by an electrical short
circuit is considered a malfunction, emergency, or accidental
load and is generally reported separately by the machinery
manufacturer. Often in the design for this phenomenon, the
magnitude of the emergency drive torque is determined by
applying a magnification factor to the normal torque.
Consultation with the generator manufacturer is necessary to
establish the appropriate magnification factor.
3.2.1.6 Special loads for elevated-type foundations—To
ensure adequate strength and deflection control, the
following special static loading conditions are recommended
in some proprietary standards for large equipment on
elevated-type foundations:
1. Vertical force equal to 50% of the total weight of each
machine;
2. Horizontal force (in the transverse direction) equal to
25% of the total weight of each machine; and
3. Horizontal force (in the longitudinal direction) equal to
25% of the total weight of each machine.
These forces are additive to normal gravity loads and are
considered to act at the centerline of the machine shaft.
Loads 1, 2, and 3 are not considered to act concurrently with
one another.
3.2.1.7 Erection and maintenance loads—Erection and
maintenance loads are temporary loads from equipment,
such as cranes and forklifts, required for installing or
dismantling machine components during erection or maintenance. Erection loads are usually furnished in the manufacturer’s foundation load drawing and should be used in
conjunction with other specified dead, live, and environmental
loads. Maintenance loads occur any time the equipment is
being drained, cleaned, repaired, and realigned or when the
components are being removed or replaced. Loads may
result from maintenance equipment, davits, and hoists. Environmental loads, such as full wind and earthquake, are not
usually assumed to act with maintenance loads, which generally occur for only a relatively short duration.
3.2.1.8 Thermal loads—Changing temperatures of
machines and their foundations cause expansions and
contractions, and distortions, causing the various parts to try
to slide on the support surfaces. The magnitude of the
resulting frictional forces depends on the magnitude of the
temperature change, the location of the supports, and on the
condition of the support surfaces. The thermal forces do not
impose a net force on the foundation to be resisted by soil or
piles because the forces on any surface are balanced by equal
and opposite forces on other support surfaces. Thermal
forces, however, may govern the design of the grout system,
pedestals, and hold downs.
Calculation of the exact thermal loading is very difficult
because it depends on a number of factors, including
distance between anchor points, magnitude of temperature
change, the material and condition of the sliding surface, and
the magnitude of the vertical load on each soleplate. Lacking
a rigorous analysis, the magnitude of the frictional load may
be calculated as follows
351.3R-9
Fig. 3.1—Equivalent forces for torque loads.
Force = (friction coefficient)(load acting through soleplate) (3-2)
The friction coefficient generally varies from 0.2 to 0.5.
Loads acting through the soleplate include: machine dead
load, normal torque load, anchor bolt load, and piping loads.
Heat transfer to the foundation can be by convection across an
air gap (for example, gap between sump and block) and by
conduction through points of physical contact. The resultant
temperature gradients induce deformations, strains, and stresses.
When evaluating thermal stress, the calculations are
strongly influenced by the stiffness and restraint against
deformation for the structural member in question. Therefore, it is important to consider the self-relieving nature of
thermal stress due to deformation to prevent being overly
conservative in the analysis. As the thermal forces are
applied to the foundation member by the machine, the foundation member changes length and thereby provides reduced
resistance to the machine forces. This phenomenon can have
the effect of reducing the thermal forces from the machine.
Accurate determinations of concrete surface temperatures
and thermal gradients are also important. Under steady-state
normal operating conditions, temperature distributions
across structural sections are usually linear. The air gap
between the machine casing and foundation provides a
significant means for dissipating heat, and its effect should
be included when establishing surface temperatures.
Normally, the expected thermal deflection at various
bearings is estimated by the manufacturer, based on past
field measurements on existing units. The machine erector
then compensates for the thermal deflection during installation.
351.3R-10
ACI COMMITTEE REPORT
Reports are available (Mandke and Smalley 1992;
Mandke and Smalley 1989; and Smalley 1985) that illustrate
the effects of thermal loads and deflections in the concrete
foundation of a large reciprocating compressor and their
influence on the machine.
3.2.2 Rotating machine loads—Typical heavy rotating
machinery include centrifugal air and gas compressors, horizontal and vertical fluid pumps, generators, rotating steam and
gas turbine drivers, centrifuges, electric motor drivers, fans,
and blowers. These types of machinery are characterized by
the rotating motion of one or more impellers or rotors.
3.2.2.1 Dynamic loads due to unbalanced masses—
Unbalanced forces in rotating machines are created when the
mass centroid of the rotating part does not coincide with the
axis of rotation. In theory, it is possible to precisely balance
the rotating elements of rotating machinery. In practice, this
is never achieved; slight mass eccentricities always remain.
During operation, the eccentric rotating mass produces
centrifugal forces that are proportional to the square of
machine speed. Centrifugal forces generally increase during
the service life of the machine due to conditions such as
machine wear, rotor play, and dirt accumulation.
A rotating machine transmits dynamic force to the foundation
predominantly through its bearings (with small, generally
unimportant exceptions such as seals and the air gap in a
motor). The forces acting at the bearings are a function of the
level and axial distribution of unbalance, the geometry of the
rotor and its bearings, the speed of rotation, and the detailed
dynamic characteristics of the rotor-bearing system. At or near
a critical speed, the force from rotating unbalance can be
substantially amplified, sometimes by a factor of five or more.
Ideally, the determination of the transmitted force under
different conditions of unbalance and at different speeds
results from a dynamic analysis of the rotor-bearing system,
using an appropriate combination of computer programs for
calculating bearing dynamic characteristics and the response
to unbalance of a flexible rotor in its bearings. Such an analysis
would usually be performed by the machine manufacturer.
Results of such analyses, especially values for transmitted
bearing forces, represent the best source of information for
use by the foundation design engineer. This and other
approaches used in practice to quantify the magnitude of
dynamic force transmitted to the foundation are discussed in
Sections 3.2.2.1a to 3.2.2.1.3e.
3.2.2.1a Dynamic load provided by the manufacturer—The engineer should request and the machine manufacturer should provide the following information:
Design levels of unbalance and basis—This information
documents the unbalance level the subsequent transmitted
forces are based on.
Dynamic forces transmitted to the bearing pedestals
under the following conditions—
a) Under design unbalance levels over operating speed
range;
b) At highest vibration when negotiating critical speeds;
c) At a vibration level where the machine is just short of
tripping on high vibration; and
d) Under the maximum level of upset condition the
machine is designed to survive (for example, loss of one or
more blades).
Items a and b document the predicted dynamic forces
resulting from levels of unbalance assumed in design for
normal operation. Using these forces, it is possible to predict
the normal dynamic vibration of the machine on its foundation.
Item c identifies a maximum level of transmitted force
with which the machine could operate continuously without
tripping; the foundation should have the strength to tolerate
such a dynamic force on a continuous basis.
Item d identifies the higher level of dynamic force, which
could occur under occasional upset conditions over a short
period of time. If the machine is designed to tolerate this
level of dynamic force for a short period of time, then the
foundation should also be able to tolerate it for a similar
period of time.
If an independent dynamic analysis of the rotor-bearing
system is performed by the end user or by a third party, such
an analysis can provide some or all of the above dynamic
forces transmitted to the foundation.
By assuming that the dynamic force transmitted to the
bearings equals the rotating unbalanced force generated by
the rotor, information on unbalance can provide an estimate
of the transmitted force.
3.2.2.1b Machine unbalance provided by the manufacturer—When the mass unbalance (eccentricity) is known or
stated by the manufacturer, the resulting dynamic force
amplitude is
Fo = mr emωo2Sf /12 lbf
(3-3)
Fo = mr emωo2Sf /1000 N
where
Fo =
mr =
em =
ωo =
dynamic force amplitude (zero-to-peak), lbf (N);
rotating mass, lbm (kg);
mass eccentricity, in. (mm);
circular operating frequency of the machine (rad/s);
and
Sf = service factor, used to account for increased
unbalance during the service life of the machine,
generally greater than or equal to 2.
3.2.2.1c Machine unbalance meeting industry
criteria—Many rotating machines are balanced to an initial
balance quality either in accordance with the manufacturer’s
procedures or as specified by the purchaser. ISO 1940 and
ASA/ANSI S2.19 define balance quality in terms of a constant
emωo. For example, the normal balance quality Q for parts of
process-plant machinery is 0.25 in./s (6.3 mm/s). Other typical
balance quality grade examples are shown in Table 3.1. To
meet these criteria a rotor intended for faster speeds should be
better balanced than one operating at a slower speed. Using
this approach, Eq. (3-3) can be rewritten as
Fo = mrQωoSf /12
lbf
Fo = mrQωoSf /1000 N
(3-4)
FOUNDATIONS FOR DYNAMIC EQUIPMENT
351.3R-11
Table 3.1—Balance quality grades for selected
groups of representative rigid rotors (excerpted
from ANSI/ASA S2.19)
Balance
quality Product of eω,
guide in./s (mm/s)
G1600
63 (1600)
Rotor types—general examples
Crankshaft/drives of rigidly mounted, large,
two-cycle engines
G630
2.5 (630)
Crankshaft/drives of rigidly mounted, large,
four-cycle engines
G250
10 (250)
Crankshaft/drives of rigidly mounted, fast,
four-cylinder diesel engines
G100
4 (100)
G40
1.6 (40)
Crankshaft/drives of fast diesel engines with six or
more cylinders
Crankshaft/drives of elastically mounted, fast
four-cycle engines (gasoline or diesel) with six or
more cylinders
G16
0.6 (16)
G6.3
0.25 (6.3)
G2.5
0.1 (2.5)
G1
0.04 (1)
G0.4
Parts of crushing machines; drive shafts (propeller
shafts, cardan shafts) with special requirements;
crankshaft/drives of engines with six or more
cylinders under special requirements
Parts of process plant machines; centrifuge drums,
paper machinery rolls, print rolls; fans; flywheels;
pump impellers; machine tool and general
machinery parts; medium and large electric
armatures (of electric motors having at least
80 mm shaft height) without special requirement
Gas and steam turbines, including marine main
turbines; rigid turbo-generator rotors; turbocompressors; machine tool drives; medium and
large electric armatures with special requirements;
turbine driven pumps
Grinding machine drives
discs, and armatures of precision
0.015 (0.4) Spindles,
grinders
API 617 and API 684 work with maximum residual unbalance Umax criteria for petroleum processing applications. The
mass eccentricity is determined by dividing Umax by the rotor
weight. For axial and centrifugal compressors with maximum
continuous operating speeds greater than 25,000 rpm, API 617
establishes a maximum allowable mass eccentricity of 10 ×
10–6 in. (250 nm). For compressors operating at slower speeds,
the maximum allowable mass eccentricity is
em = 0.25/fo
em = 6.35/fo
in.
(3-5)
mm
where
fo = operating speed, rpm ≤ 25,000 rpm.
This permitted initial mass eccentricity is tighter than ISO
balance quality grade G2.5, which would be applied to this
type of equipment (Table 3.1, turbo-compressors) under ISO
1940. As such, the dynamic force computed from this API
consideration will be quite small and a larger service factor
might be used to have a realistic design force.
API 617 also identifies a limitation on the peak-to-peak
vibration amplitude during mechanical testing of the
compressor with the equipment operating at its maximum
continuous speed ((12,000/fo)0.5 in. [25.4(12,000/fo)0.5 mm]).
Some design firms use this criterion and a service factor Sf
of 2.0 to compute the dynamic force amplitude as
Fig. 3.2—Comparison of effective eccentricity.
W r f o1.5
F o = -----------------322,000
(3-6)
where Wr = rotating weight, lbf (N).
3.2.2.1d Dynamic load determined from an empirical
formula—Rotating machine manufacturers often do not
report the unbalance that remains after balancing. Consequently, empirical formulas are frequently used to ensure
that foundations are designed for some minimum unbalance,
which generally includes some allowance for increasing
unbalance over time. One general purpose empirical method
assumes that balancing improves with machine speed and
that there is a linear relationship between the unbalanced
forces and the machine speed. The zero-to-peak centrifugal
force amplitude from one such commonly used expression is
Wr fo
F o = ----------6000
(3-7)
Equations (3-3), (3-4), (3-6), and (3-7) appear to be very
different: the exponents on the speed of rotation vary from 1
to 1.5 to 2, constants vary widely, and different variables
appear. Some equations use mass, others use weight. In
reality, the equations are more similar than they appear.
Given the right understanding of Q as a replacement for eω,
Eq. (3-3), (3-4), and (3-7) take on the same character. These
equations then indicate that the design force at operating
speed varies linearly with both the mass of the rotating body
and the operating rotational speed. Once that state is identified,
Eq. (3-3) can be adjusted to reflect the actual speed of rotation,
and the dynamic centrifugal force is seen to vary with the
square of the speed. Restating Eq. (3-6) and (3-7) in the form
of Eq. (3-3) allows for the development of an effective eccentricity implied within these equations with the comparison
shown in Fig. 3.2. Equation (3-7) produces the same result as
Eq. (3-4) using Q = 0.25 in./s (6.3 mm/s), and Sf = 2.5.
The centrifugal forces due to mass unbalance are considered
to act at the center of gravity of the rotating part and vary
harmonically at the speed of the machine in the two orthogonal
directions perpendicular to the shaft. The forces in the two
orthogonal directions are equal in magnitude and 90 degrees
out of phase and are transmitted to the foundation through the
351.3R-12
ACI COMMITTEE REPORT
Fig. 3.3—Crack mechanism.
bearings. Schenck (1990) provides useful information about
balance quality for various classes of machinery.
3.2.2.1e Machine unbalance determined from trip
vibration level and effective bearing stiffness—Because a
rotor is often set to trip on high vibration, it can be expected
to operate continuously at any vibration level up to the trip
limit. Given the effective bearing stiffness, it is possible to
calculate the maximum dynamic force amplitude as
Fo = Vmax Keff
(3-8)
where
Vmax = the maximum allowable vibration, in. (mm); and
Keff = the effective bearing stiffness, lbf/in. (N/mm).
To use this approach, the manufacturer should provide
effective bearing stiffness or the engineer should calculate it
from the bearing geometry and operating conditions (such as
viscosity and speed).
3.2.2.2 Loads from multiple rotating machines—If a
foundation supports multiple rotating machines, the engineer
should compute unbalanced force based on the mass, unbalance, and operating speed of each rotating component. The
response to each rotating mass is then combined to determine
the total response. Some practitioners, depending on the
specific situation of machine size and criticality, find it
advantageous to combine the unbalanced forces from each
rotating component into a single resultant unbalanced force.
The method of combining two dynamic forces is up to individual judgment and often involves some approximations. In
some cases, loads or responses can be added absolutely. In
other cases, the loads are treated as out-of-phase so that
twisting effects are increased. Often, the operating speed of
the equipment should be considered. Even if operating
speeds are nominally the same, the design engineer should
recognize that during normal operation, the speed of the
machines will vary and beating effects can develop. Beating
effects develop as two machines operate at close to the same
speed. At one point in time, responses to the two machines are
additive and motions are maximized. A short time later, the
responses cancel each other and the motions are minimized.
The net effect is a continual cyclic rising and falling of motion.
3.2.3 Reciprocating machine loads—Internal-combustion
engines, piston-type compressors and pumps, some metal
forming presses, steam engines, and other machinery are
characterized by the rotating motion of a master crankshaft
and the linear reciprocating motion of connected pistons or
sliders. The motion of these components cause cyclically
varying forces, often called reciprocating forces.
3.2.3.1 Primary and secondary reciprocating loads—
The simplest type of reciprocating machine uses a single
crank mechanism as shown in Fig. 3.3. The idealization of
this mechanism consists of a piston that moves within a
guiding cylinder, a crank of length r that rotates about a
crank shaft, and a connecting rod of length L. The connecting
rod is attached to the piston at point P and to the crank at
point C. The wrist pin P oscillates while the crank pin C
follows a circular path. This idealized single cylinder illustrates the concept of a machine producing both primary and
secondary reciprocating forces.
If the crank is assumed to rotate at a constant angular
velocity ωo, the translational acceleration of the piston along
its axis may be evaluated. If Zp is defined as the piston
displacement toward the crankshaft (local Z-axis), an
expression can be written for Zp at any time t. Further, the
velocity and acceleration can also be obtained by taking the first
and second derivatives of the displacement expression with
respect to time. The displacement, velocity, and acceleration
expressions for the motion of the piston are as follows
2
r
r
Z p =  r + ------ – r  cos ω o t + ------ cos 2ω o t



4L
4L
(3-9)
r
Z· p = rω o  sin ω o t + ------ sin 2ω o t


2L
(3-10)
..
2
r
Zp = rω o  cos ω o t --- cos 2ω o t


L
(3-11)
where
Zp = piston displacement, in. (mm);
r
= length of crank, in. (mm);
L = length of connecting rod, in. (mm);
ωo = circular operating frequency of the machine (rad/s); and
t
= time, s.
Note that the expressions contain two terms each with a
sine or cosine; the term that varies with the frequency of the
rotation, ωo, is referred to as the primary term while the term
that varies at twice the frequency of rotation, 2ωo, is called
the secondary term.
Similar expressions can be developed for the local Z-axis
(parallel to piston movement) and local Y-axis (perpendicular to piston movement) motion of the rotating parts of the
crank. If any unbalance in the crankshaft is replaced by a
mass concentrated at the crank pin C, such that the inertia
forces are the same as in the original system, the following
terms for motion at point C can be written
FOUNDATIONS FOR DYNAMIC EQUIPMENT
yp = –r sinωot
(3-12)
·
y p = –r ωcosωot
(3-13)
··
y p = r ωo2sinωot
(3-14)
zp = r(1 – cosωot)
(3-15)
·
z p = rωosinωot
(3-16)
··
z p = r ωo2cosωot
(3-17)
where
yp = crank pin displacement in local Y-axis, in. (mm); and
zp = crank pin displacement in local Z-axis, in. (mm).
Identifying a part of the connecting rod (usually 1/3 of its
mass) plus the piston as the reciprocating mass mrec concentrated at point P and designating the remainder of the
connecting rod plus the crank as the rotating mass mrot
concentrated at point C, expressions for the unbalanced
forces are as follows
351.3R-13
Fig. 3.4—Schematic of double-acting compressor cylinder
and piston.
The gas force contributed to the piston rod equals the
instantaneous difference between the pressure force acting
on the head and crank end of the piston as shown in Fig. 3.4.
The following formulation can be used to estimate the
maximum force acting on the piston rod of an individual
double-acting cylinder
Frod = [(Phead)(Ahead) – (Pcrank)(Acrank)] F1
(3-22)
Ahead = (π/4)Bc2
(3-23)
Acrank = (π/4)(Bc2 – Drod2)
(3-24)
Parallel to piston movement
2
2
r ω
2
F z = ( m rec + m rot )rω o cos ω o t + m rec ----------o- cos 2ω o t (3-18)
L
Perpendicular to piston movement
2
F Y = m rot rω o sin ω o t
(3-19)
Note that Eq. (3.18) consists of two terms, a primary force
(mrec + mrot )rωo2cosωo t
(3-20)
and a secondary force
2
2
r ω
m rec = ----------o- cos 2ω o t
L
(3-21)
whereas Eq. (3-19) has only a primary component.
3.2.3.2 Compressor gas loads—A reciprocating
compressor raises the pressure of a certain flow of gas by
imparting reciprocating motion on a piston within a cylinder.
The piston normally compresses gas during both directions
of reciprocating motion. As gas flows to and from each end,
the pressure of the gas increases as it is compressed by each
stroke of the piston. The increase in pressure within the
cylinder creates reaction forces on the head and crank ends
of the piston which alternate as gas flows to and from each
end of the cylinder.
where
Frod
Ahead,
Acrank
Bc
Drod
Phead,
Pcrank
=
force acting on piston rod, lbf (N);
=
=
=
head and crank areas, in.2 (mm2);
cylinder bore diameter, in. (mm);
rod diameter, in. (mm);
=
instantaneous head and crank pressures, psi,
(MPa); and
F1
= correction factor.
The head and crank end pressures vary continuously and
the differential force takes both positive and negative net
values during each cycle of piston motion. The normal
approach is to establish the head and crank pressures using
the maximum and minimum suction and discharge pressures.
For design purposes, it is common to multiply Eq. (3-22) by a
factor F1 to help account for the natural tendency of gas forces
to exceed the values based directly on suction and discharge
pressures due to flow resistances and pulsations. Machines
with good pulsation control and low external flow resistance
may achieve F1 as small as 1.1; for machines with low
compression ratio, high pulsations, or highly resistive flow
through piping and nozzles, F1 can approach 1.5 or even
higher. A reasonable working value for F1 is 1.15 to 1.2.
Preferably, the maximum rod force resulting from gas
pressures is based on knowledge of the continuous variation of
pressure in the cylinder (measured or predicted). In a repair
situation, measured cylinder pressure variation using a cylinder
analyzer provides the most accurate value of gas forces. Even
without cylinder pressure analysis, extreme operating values of
351.3R-14
ACI COMMITTEE REPORT
suction and discharge pressure for each stage should be
recorded before the repair and used in the Eq. (3-22).
On new compressors, the engineer should ask the machine
manufacturer to provide values for maximum compressive
and tensile gas loads on each cylinder rod and, if these are
based on suction and discharge pressures, to recommend a
value of F1.
Gas forces act on the crankshaft with an equal and opposite
reaction on the cylinder. Thus, crankshaft and cylinder
forces globally balance each other. Between the crankshaft
and the cylinder, however, the compressor frame stretches or
contracts in tension or compression under the action of the
gas forces. The forces due to frame deflections are transmitted to the foundation through connections with the
compressor frame. When acting without slippage, the frame
and foundation become an integral structure and together
stretch or contract under the gas loads.
The magnitude of gas force transferred into the foundation
depends on the relative flexibility of the compressor frame.
A very stiff frame transmits only a small fraction of the gas
force while a very flexible frame transmits most or all of the
force. Similar comments apply to the transfer of individual
cylinder inertia forces.
Based on limited comparisons using finite element analysis
(Smalley 1988), the following guideline is suggested for gas
and inertia force loads transmitted to the foundation by a
typical compressor
where
Fblock
Fblock = Frod /Fred
(3-25)
(Fbolt)CHG = [(Frod)/(Nbolt)CHG]/Fred
(3-26)
(Fbolt)frame = [(Funbalance/(Nbolt)]/Fred
(3-27)
=
the force acting outward on the block from
which concrete stresses should be calculated, lbf (N);
(Fbolt)CHG = the force to be restrained by friction at the
cross head guide tie-down bolts, lbf (N);
(Fbolt)frame = the force to be restrained by friction at the
frame tie-down bolts, lbf (N);
Fred
= a force reduction factor with suggested
value of 2, to account for the fraction of
individual cylinder load carried by the
compressor frame (“frame rigidity factor”);
(Nbolt)CHG = the number of bolts holding down one
crosshead guide;
(Nbolt)frame = the number of bolts holding down the
frame, per cylinder;
Frod
= force acting on piston rod, from Eq. (3-22),
lbf (N); and
Funbalance = the maximum value from Eq. (3-18) applied
using parameters for a horizontal
compressor cylinder, lbf (N).
The factor Fred is used to simplify a complex problem, thus
avoiding the application of unrealistically high loads on the
anchor bolts and the foundation block. The mechanics
involved in transmitting loads are complex and cannot easily
be reduced to a simple relationship between a few parameters
beyond the given load equations. A detailed finite-element
analysis of metal compressor frame, chock mounts, concrete
block, and grout will account for the relative flexibility of the
frame and its foundation in determining individual anchor
bolt loads and implicitly provide a value for Fred. If the
frame is very stiff relative to the foundation, the value for
Fred will be higher, implying more of the transmitted loads
are carried by the frame and less by the anchor bolts and
foundation block. Based on experience, a value of 2 for this
factor is conservatively low; however, higher values have
been seen with frames designed to be especially stiff.
Simplifying this approach, one report (Smalley and
Harrell 1997) suggests using a finite element analysis to
calculate forces transmitted to the anchor bolts. If a finite
element analysis is not possible, the engineer should get
from the machine manufacturer or calculate the maximum
horizontal gas force and maximum horizontal inertia force
for any throw or cylinder. The mounts, anchor bolts, and
blocks are then designed for
FTHROW = (greater of FGMAX or FIMAX)/2
where
FGMAX
(3-28)
= maximum horizontal gas force on a throw or
cylinder, lbf (N);
= maximum horizontal inertia force on a throw or
FIMAX
cylinder, lbf (N); and
FTHROW = horizontal force to be resisted by each throw’s
anchor bolts, lbf (N).
3.2.3.3 Reciprocating inertia loads for multicylinder
machines—As a practical matter, most reciprocating
machines have more than one cylinder, and manufacturers
arrange the machine components in a manner that minimizes
the net unbalanced forces. For example, rotating parts like
the crankshaft can be balanced by adding or removing
correcting weights. Translating parts like pistons and those
that exhibit both rotation and translation, like connecting
rods, can be arranged in such a way as to minimize the unbalanced forces and moments generated. Seldom, if ever, is it
possible to perfectly balance reciprocating machines.
The forces generated by reciprocating mechanisms are
functions of the mass, stroke, piston arrangement,
connecting rod size, crank throw orientation (phase angle),
and the mass and arrangement of counterweights on the
crankshaft. For this reason, calculating the reciprocating
forces for multicylinder machines can be quite complex and
are therefore normally provided by the machine manufacturer. If the machine is an integral engine compressor, it can
include, in one frame, cylinders oriented horizontally, vertically, or in between, all with reciprocating inertias.
Some machine manufacturers place displacement transducers
and accelerometers on strategic points on the machinery. They
can then measure displacements and accelerations at those
points for several operational frequencies to determine the
FOUNDATIONS FOR DYNAMIC EQUIPMENT
magnitude of the unbalanced forces and couples for multicylinder machines.
3.2.3.4 Estimating reciprocating inertia forces from
multicylinder machines—In cases where the manufacturer’s
data are unavailable or components are being replaced, the
engineer should use hand calculations to estimate the reciprocating forces from a multicylinder machine. One such
procedure for a machine having n number of cylinders is
discussed by Mandke and Troxler (1992). Section 3.7
summarizes this method.
3.2.4 Impulsive machine loads—The impulsive load
generated by a forging hammer is caused by the impact of the
hammer ram onto the hammer anvil. This impact process
transfers the kinetic energy of the ram into kinetic energy of
the entire hammer assembly. The post-impact velocity of the
hammer is represented by
M
v h = -------r ( 1 + α h )v r
Mh
(3-29)
where
vh = post-impact hammer velocity, ft/s (m/s);
Mr = ram mass including dies and ancillary parts, lbm (kg);
Mh = hammer mass including any auxiliary foundation, lbm
(kg);
αh = ram rebound velocity relative to impact velocity; and
vr = ram impact velocity, ft/s (m/s).
General experience indicates that αh is approximately
60% for many forging hammer installations. From that point,
the hammer foundation performance can be assessed as a
rigid body oscillating as a single degree-of-freedom system
with an initial velocity of vh.
For metal-forming presses, the dynamic forces develop
from two sources: the mechanical movement of the press
components and material-forming process. Each of these
forces is unique to the press design and application and needs
to be evaluated with proper information from the press
manufacturer and the owner.
The press mechanics often include rotating and reciprocating components. The dynamic forces from these individual pieces follow the rules established in earlier sections
of this document for rotating and reciprocating components.
Only the press manufacturer familiar with all the internal
components can knowledgeably calculate the specific
forces. Figure 2.3 presents a horizontal force time-history for
a forging press. Similar presses can be expected to have
similar characteristics; however, the particular values and
timing data differ.
The press drive mechanisms include geared and directdrive systems. Depending on the design, these drives may or
may not be balanced. The press slide travels vertically
through a set stroke of 1/2 in. (12 mm) to several inches at a
given speed. Some small presses may have inclinable beds
so that the slide is not moving vertically. It is often adequate
to assume that the slide moves in a vertical path defined by a
circularly rotating crankshaft, that is
351.3R-15
S
d s ( t ) = --- sin ( ω o t )
2
(3-30)
where
ds = displacement of the slide, in. (mm);
S = press stroke, in. (mm); and
ωo = circular operating frequency of the machine (rad/s).
This leads to a dynamic inertia force from the slide of
2S
F s ( t ) = m d ω o --- sin ( ω o t ) ⁄ 12 lbf
2
(3-31)
F s ( t ) = m d ω o S--- sin ( ω o t ) ⁄ 1000 N
2
2
where
Fs = dynamic inertia force of slide, lbf (N); and
md = slide mass including the effects of any balance mechanism, lbm (kg).
This assumption is based on simple circular motions and
simple linkages. Other systems may be in-place to increase
the press force and improve the timing. These other systems
may increase the acceleration of the unbalanced weights and
thus alter the magnitude and frequency components of the
dynamic force transmitted to the foundation.
3.2.5 Loading conditions—During their lives, machinery
equipment support structures and foundations undergo
different loading conditions including erection, testing, shutdown, maintenance, and normal and abnormal operation. For
each loading condition, there can be one or more combinations
of loads that apply to the structure or foundation. The following
loading conditions are generally considered in design:
• Erection condition represents the design loads that act
on the structure/foundation during its construction;
• Testing condition represents the design loads that act on
the structure/foundation while the equipment being
supported is undergoing testing, such as hydrotest;
• Empty (shutdown) represents the design loads that act
on the structure when the supported equipment is at its
least weight due to removal of process fluids, applicable
internals, or both as a result of maintenance or other
out-of-service disruption;
• Normal operating condition represents the design loading
during periods of normal equipment operation; and
• Abnormal operating condition represents the design
loading during periods when unusual or extreme operating
loads act on the structure/foundation.
3.2.6 Load combinations—Table 3.2 shows the general
classification of loads for use in determining the applicable
load factors in strength design (ACI 318). In considering soil
stresses, the normal approach is working stress design
without load factors and with overall factors of safety identified as appropriate by geotechnical engineers. The load
combinations frequently used for the various load conditions
are as follows:
1. Erection
a) Dead load + erection forces
351.3R-16
ACI COMMITTEE REPORT
Table 3.2—Load classifications for ultimate
strength design
Design loads
Weight of structure, equipment, internals, insulation, and
platforms
Temporary loads and forces caused by erection
Fluid loads during testing and operation
Thermal loads
Anchor and guide loads
Platform and walkway loads
Materials to be temporarily stored during maintenance
Materials normally stored during operation such as tools
and maintenance equipment
Vibrating equipment forces
Impact loads for hoist and equipment handling utilities
Earthquake loads
Transportation loads
Snow, ice, or rain loads
Wind loads
Load
classification
Dead
Live
Environmental
b) Dead load + erection forces + reduced wind + snow,
ice, or rain
c) Dead load + erection forces + seismic + snow, ice, or rain
2. Testing
a) Dead load + test loads
b) Dead load + test loads + live + snow, ice, or rain
c) Dead load + test loads + reduced wind + snow, ice,
or rain
3. Empty (shutdown)
a) Dead load + maintenance forces + live load + snow,
ice, or rain
4. Normal operation
a) Dead load
b) Dead load + thermal load + machine forces + live
loads + wind + snow, ice, or rain
c) Dead load + thermal load + machine forces + seismic
+ snow, ice, or rain
5) Abnormal operation
a) Dead load + upset (abnormal) machine loads + live +
reduced wind
It is common to only use some fraction of full wind, such as
80% in combination with erection loads and 33% for test
loads, due to the short duration of these conditions (ASCE 7).
3.3—Dynamic soil properties
Soil dynamics deals with engineering properties and
behavior of soil under dynamic stress. For the dynamic analysis
of machine foundations, soil properties, such as Poisson’s
ratio, dynamic shear modulus, and damping of soil, are
generally required.
Though this work is typically completed by a geotechnical
engineer, this section provides a general overview of
methods used to determine the various soil properties. Many
references are available that provide a greater level of detail
on both theory and standard practice, including Das (1993),
Bowles (1996), Fang (1991), and Arya, O’Neill, and Pincus
(1979). Seed and Idriss (1970) provide greater detail on
items that influence different soil properties.
This section does not cover considerations that affect the
suitability of a given soil to support a dynamic machine
foundation. Problems could include excessive settlement
caused by dynamic or static loads, liquefaction, dimensional
stability of a cohesive soil, frost heave, or any other relevant
soils concern.
In general, problems involving the dynamic properties of
soils are divided into small and large strain amplitude
responses. For machine foundations, the amplitudes of
dynamic motion, and consequently the strains in the soil, are
usually low (strains less than10–3%). A foundation that is
subjected to an earthquake or blast loading is likely to
undergo large deformations and, therefore, induce large
strains in the soil. The information in this report is only
applicable for typical machine foundation strains. Refer to
Seed and Idriss (1970) for information on strain-related
effects on shear modulus and material damping.
The key soil properties, Poisson’s ratio and dynamic shear
modulus, may be significantly affected by water table variations. Prudence suggests that in determining these properties,
such variations be considered and assessed, usually in
conjunction with the geotechnical engineers. This approach
often results in expanding the range of properties to be
considered in the design phase.
3.3.1 Poisson’s ratio—Poisson’s ratio ν, which is the ratio
of the strain in the direction perpendicular to loading to the
strain in the direction of loading, is used to calculate both the
soil stiffness and damping. Poisson’s ratio can be computed
from the measured values of wave velocities traveling through
the soil. These computations, however, are difficult. The
stiffness and damping of a foundation system are generally
insensitive to variations of Poisson’s ratio common in soils.
Generally, Poisson’s ratio varies from 0.25 to 0.35 for
cohesionless soils and from 0.35 to 0.45 for cohesive soils. If
no specific values of Poisson’s ratio are available, then, for
design purposes, the engineer may take Poisson’s ratio as
0.33 for cohesionless soils and 0.40 for cohesive soils.
3.3.2 Dynamic shear modulus—Dynamic shear modulus
G is the most important soil parameter influencing the
dynamic behavior of the soil-foundation system. Together
with Poisson’s ratio, it is used to calculate soil impedance.
Refer to Section 4.2 for the discussion on soil impedance.
The dynamic shear modulus represents the slope of the
shear stress versus shear strain curve. Most soils do not
respond elastically to shear strains; they respond with a
combination of elastic and plastic strain. For that reason,
plotting shear stress versus shear strain results in a curve not
a straight line. The value of G varies based on the strain
considered. The lower the strain, the higher the dynamic
shear modulus.
Several methods are available for obtaining useful values
of dynamic shear modulus:
• Field measurements of stress wave velocities of in-place
soils;
• Laboratory tests on soil samples; and
• Correlation to other soil properties.
Due to variations inherent in the determination of dynamic
shear modulus values, it may be appropriate to complete more
than one foundation analysis. One analysis could be
completed with the minimum possible value, one could be
completed using the maximum possible value, and then additional analyses could be completed with intermediate values.
FOUNDATIONS FOR DYNAMIC EQUIPMENT
3.3.2.1 Field determination—Field measurements are
the most common method for determining the dynamic shear
modulus of a given soil. These methods involve measuring
the soil characteristics, in-place, as close as possible to the
actual foundation location(s).
Because field determinations are an indirect determination
of shear modulus, the specific property measured is the shear
wave velocity. There are three different types of stress waves
that can be transmitted through soil or any other elastic body.
• Compression (primary P) waves;
• Shear (secondary S) waves; and
• Rayleigh (surface) waves.
Compression waves are transmitted through soil by a
volume change associated with compressive and tensile
stresses. Compression waves are the fastest of the three
stress waves.
Shear waves are transmitted through soil by distortion
associated with shear stresses in the soil and are slower than
compression waves. No volume change occurs in the soil.
Rayleigh waves occur at the free surface of an elastic body;
typically, this is the ground surface. Rayleigh waves have
components that are both perpendicular to the free surface
and parallel to the free surface and are slightly slower than
shear waves.
Several methods are available for measuring wave velocities of the in-place soil:
• The cross-hole method;
• The down-hole method;
• The up-hole method; and
• Seismic reflection (or refraction).
In the cross-hole method, two vertical boreholes are
drilled. A signal generator is placed in one hole and a sensor
is placed in the other hole. An impulse signal is generated in
one hole, and then the time the shear wave takes to travel
from the signal generator to the sensor is measured. The
travel time divided by the distance yields the shear wave
velocity. The cross-hole method can be used to determine G
at different depths (Fig. 3.5).
In the down-hole method, only one vertical borehole is
drilled. A signal generator is placed at the ground surface
some distance away from the borehole, and a sensor is placed
in the bottom of the borehole. An impulse signal is generated,
and then the time the shear wave takes to travel from the
signal generator to the sensor is measured. The travel time
divided by the distance yields the shear wave velocity. This
method can be run several different times, with the signal
generator located at different distances from the borehole
each time. This permits the measuring of soil properties at
several different locations, which can then be averaged to
determine an average shear wave velocity (Fig. 3.6).
The up-hole method is similar to the down-hole method.
The difference is that the signal generator is placed in the
borehole and the sensor is placed at the ground surface.
Dynamic shear modulus and measured-in-field shear
wave velocity are related as follows
G = ρ(Vs)2
(3-32)
351.3R-17
Fig. 3.5—Schematic of cross-hole technique.
Fig. 3.6—Equipment and instrumentation for down-hole
survey.
where
G = dynamic shear modulus of the soil, lbf/ft2 (Pa);
Vs = shear wave velocity of the soil, ft/s (m/s); and
ρ = soil mass density, lbm/ft3 (kg/m3).
An alternative field method is to use reflection or refraction
of elastic stress waves. These methods are based on the
principle that when elastic waves hit a boundary between
dissimilar layers, the wave is reflected or refracted. This
method should only be used at locations where the soils are
deposited in discrete horizontal, or nearly horizontal layers,
or at locations where soil exists over top of bedrock. This
method consists of generating a stress wave at one location
at the ground surface and measuring the time it takes for the
stress wave to reach a second location at the ground surface.
The wave travels from the ground surface to the interface
between differing soils layers, travels along the interface,
then back to the ground surface. The time the wave takes to
travel from the signal generator to the sensor is a function of
the soils properties and the depth of the soil interface. One
advantage of this method is that no boreholes are required.
Also, this method yields an estimated depth to differing soil
layers. One disadvantage is that this method cannot be used
when the groundwater table is near the ground surface.
3.3.2.2 Laboratory determination—Laboratory tests are
considered less accurate than field measurements due to the
351.3R-18
ACI COMMITTEE REPORT
Table 3.3—Values of K2 versus relative density
(Seed and Idriss 1970)
Relative density, %
K2
90
70
75
60
61
52
45
40
43
40
30
34
G = 1000K 2 σ o lb/ft2
possibility of sample disturbance. Sometimes laboratory
tests are used to validate field measurements when a high
level of scrutiny is required, for instance, when soil properties
are required for a nuclear energy facility.
The most common laboratory test is the Resonant-Column
method, where a cylindrical sample of soil is placed in a
device capable of generating forced vibrations. The soil
sample is exited at different frequencies until the resonant
frequency is determined. The dynamic soil modulus can be
calculated based on the frequency, the length of the soil
sample, the end conditions of the soil sample, and the density
of the soil sample. ASTM D 4015 defines the ResonantColumn method.
3.3.2.3 Correlation to other soil properties—Correlation
is another method for determining dynamic soils properties.
The engineer should be careful when using any correlation
method because these are generally the least-accurate
methods. The most appropriate time to consider using these
methods is for preliminary design or for small noncritical
applications with small dynamic loads. Correlation to other
soil properties should be considered as providing a range of
possible values, not providing a single exact value.
Hardin and Richart (1963) determined that soil void ratio
ev and the probable confining pressure σo had the most
impact on the dynamic shear modulus. Hardin and Black
(1968) developed the following relationships:
For round-grained sands with e < 0.8, dynamic shear
modulus can be estimated from
2
31,530 ( 2.17 – e v ) σ o
lbf/ft2
G = ------------------------------------------------------1 + ev
(3-33)
For angular-grained materials with e > 0.6 and normally
consolidated clays with low surface activity, dynamic shear
modulus can be estimated from
2
2
102,140 ( 2.97 – e v ) σ o
G = ---------------------------------------------------------Pa
1 + ev
(3-34)
(3-35)
G = 6920K 2 σ o Pa
where K2 = a parameter that depends on void ratio and strain
amplitude. Table 3.3 provides values of K2 with respect to
relative density.
3.3.3 Damping of soil—Damping is a phenomenon of
energy dissipation that opposes free vibrations of a system.
Like the restoring forces, the damping forces oppose the
motion, but the energy dissipated through damping cannot
be recovered. A characteristic feature of damping forces is
that they lag the displacement and are out of phase with the
motion. Damping of soil includes two effects—geometric
and material damping.
Geometric, or radiation, damping reflects energy dissipation
through propagation of elastic waves away from the immediate
vicinity of a foundation and inelastic deformation of soil. It
results from the practical infinity of the soil medium, and it
is close to viscous in character. Refer to Chapter 4 for
methods of computing geometric damping.
Material, or hysteretic, damping reflects energy dissipation
within the soil itself due to the imperfect elasticity of real
materials, which exhibit a hysteric loop effect under cyclic
loading (Fig. 3.7). The amount of dissipated energy is given
by the area of the hysteretic loop. The hysteretic loop implies
a phase shift between the stress and strain because there is a
stress at zero strain and vice versa, as can be seen from Fig. 3.7.
The amount of dissipated energy depends on strain
(displacement) but is essentially independent of frequency,
as shown on Fig. 3.8.
The magnitude of material damping can be established
experimentally using the hysteretic loop and the relation
1 ∆W
β m = ------ --------4π W
2
218,200 ( 2.17 – e v ) σ o
G = ---------------------------------------------------------Pa
1 + ev
14,760 ( 2.97 – e v ) σ o
lbf/ft2
G = ------------------------------------------------------1 + ev
In the previous equations,
ev = void ratio; and
σo = probable confining pressure, lbf/ft2 (Pa).
In general, relative density in sand is proportional to the
void ratio. Seed and Idriss (1970) provide guidance for
correlating the dynamic shear modulus to relative density in
sand, along with the confining pressure
(3-36)
where
βm = material damping ratio;
∆W = area enclosed by the hysteretic loop; and
W
= strain energy.
Instead of an experimental determination, many practitioners
use a material damping ratio of 0.05, or 5%. The material
damping ratio is fairly constant for small strains but increases
with strain due to the nonlinear behavior of soils.
The term material or hysteretic damping implies
frequency independent damping. Experiments indicate that
frequency independent hysteretic damping is much more
FOUNDATIONS FOR DYNAMIC EQUIPMENT
351.3R-19
typical of soils than viscous damping because the area of the
hysteretic loop does not grow in proportion to the frequency.
3.4—Vibration performance criteria
The main purposes of the foundation system with respect
to dynamic loads include limiting vibrations, internal loads,
and stresses within the equipment. The foundation system
also limits vibrations in the areas around the equipment
where other vibration-sensitive equipment may be installed,
personnel may have to work on a regular basis, or damage to
the surrounding structures may occur. These performance
criteria are usually established based on vibration amplitudes
at key points on or around the equipment and foundation
system. These amplitudes may be based on displacement,
velocity, or acceleration units. Displacement limitations are
commonly based on peak-to-peak amplitudes measured in
mils (0.001 in.) or microns (10–6 m). Velocity limitations are
typically based on either peak velocities or root-mean-square
(rms) velocities in units of inch per second or millimeter per
second. Displacement criteria are almost always frequency
dependent with greater motions tolerated at slower speeds.
Velocity criteria may depend on frequency but are often
independent. Acceleration criteria may be constant with
frequency or dependent.
Some types of equipment operate at a constant speed while
other types operate across a range of speeds. The foundation
engineer should consider the effect of these speed variations
during the foundation design.
3.4.1 Machine limits—The vibration limits applicable to
the machine are normally set by the equipment manufacturer
or are specified by the equipment operator or owner. The
limits are usually predicated on either limiting damage to the
equipment or ensuring proper performance of the equipment.
Limits specified by operators of the machinery and design
engineers are usually based on such factors as experience or
the installation of additional vibration monitoring equipment.
For rotating equipment (fans, pumps, and turbines), the
normal criterion limits vibration displacements or velocities at
the bearings of the rotating shaft. Excessive vibrations of the
bearings increase maintenance requirements and lead to
premature failure of the bearings. Often, rotating equipment
has vibration switches to stop the equipment if vibrations
become excessive.
Reciprocating equipment (diesel generators, compressors,
and similar machinery) tends to be more dynamically rugged
than rotating equipment. At the same time, it often generates
greater dynamic forces. While the limits may be higher,
motions are measured at bearing locations. In addition, operators of reciprocating compressors often monitor vibrations of
the compressor base relative to the foundation (sometimes
called “frame movement”) as a measure of the foundation and
machine-mounting condition and integrity.
Impulsive machines (presses, forging hammers) tend not to
have specific vibration limitations as controllable by the foundation design. With these machines, it is important to recognize
the difference between the inertial forces and equipment
dynamics as contrasted with the foundation system dynamics.
The forces with the equipment can generate significant
Fig. 3.7—Hysteretic loop.
Fig. 3.8—Comparison of viscous and hysteretic damping.
accelerations and stresses that are unrelated to the stiffness,
mass, or other design aspect of the foundation system. Thus,
monitoring accelerations in particular on an equipment frame
may not be indicative of foundation suitability or adequacy.
Researchers have presented various studies and papers
addressing the issues of machinery vibration limits. This
variety is reflected in the standards of engineering companies,
plant owners, and industry standards. When the equipment
manufacturer does not establish limits, recommendation from
ISO 10816-1, Blake (1964), and Baxter and Bernhard (1967)
are often followed. Most of these studies relate directly to
rotating equipment. In many cases they are also applicable to
reciprocating equipment. Rarely do these studies apply to
impulsive equipment.
ISO publishes ISO 10816 in a series of six parts to address
evaluation of machinery vibration by measurements on the
nonrotating parts. Part 1 provides general guidelines and sets
the overall rules with the subsequent parts providing specific
values for specific machinery types. These standards are
primarily directed toward in-place measurements for the
assessment of machinery operation. They are not intended to
identify design standards. Design engineers, however, have
used predecessor documents to ISO 10816 as a baseline for
design calculations and can be expected to do similarly with
these more recent standards.
The document presents vibration criteria in terms of rms
velocity. Where there is complexity in the vibration signal
(beyond simple rotor unbalance), the rms velocity basis
provides the broad measure of vibration severity and can be
correlated to likely machine damage. For situations where
the pattern of motion is fairly characterized by one simple
harmonic, such as simple rotor unbalance, the rms velocities
can be multiplied by √2 to determine corresponding peak
velocity criteria. For these same cases, displacements can be
calculated as
351.3R-20
ACI COMMITTEE REPORT
Table 3.4—Service factors from Blake (Richart,
Hall, and Woods 1970)
Bolted
down
Not
bolted
down
Single-stage centrifugal pump, electric motor, fan
Typical chemical processing equipment, noncritical
1.0
1.0
0.4
0.4
Turbine, turbo-generator, centrifugal compressor
Centrifugal, stiff-shaft (at basket housing), multi-stage
centrifugal pump
Miscellaneous equipment, characteristics unknown
1.6
0.6
2.0
0.8
2.0
0.8
Centrifuge, shaft-suspended, on shaft near basket
Centrifuge, link-suspended, slung
0.5
0.3
0.2
0.1
Item
Notes: 1. Vibration is measured at the bearing housing except as noted; 2. Machine
tools are excluded; and 3) Compared or measured displacements are multiplied by the
appropriate service factor before comparing with Fig. 3.9.
Fig. 3.9—Vibration criteria for rotating machinery (Blake
1964, as modified by Arya, O’Neill, and Pincus 1979).
v′
v′
v = ------- = ----------ωm
2πf m
(3-37)
where
v = displacement amplitude, in. (cm);
v′ = velocity, in./s (cm/s);
ωm = circular frequency of motion, rad/s; and
fm = frequency of motion, Hz.
The rms velocity yields an rms displacement, and a peak
velocity results in a zero-to-peak displacement value, which
can be doubled to determine a peak-to-peak displacement
value. If the motion is not a simple pure harmonic motion, a
simple relationship among the rms displacement, rms velocity,
peak velocity, zero-to-peak displacement, and peak-to-peak
displacement does not exist.
ISO 10816-1 identifies four areas of interest with respect
to the magnitude of vibration measured:
• Zone A: vibration typical of new equipment;
• Zone B: vibration normally considered acceptable for
long-term operation;
• Zone C: vibration normally considered unsatisfactory
for long-term operation; and
• Zone D: vibration normally considered severe enough
to damage the machine.
The subsequent parts of ISO 10816 establish the boundaries
between these zones as applicable to specific equipment.
Part 2, ISO 10816-2, establishes criteria for large, landbased, steam-turbine generator sets rated over 67,000 horsepower (50 MW). The most general of the standards is Part 3,
ISO 10816-3, which addresses in-place evaluation of general
industrial machinery nominally more than 15 kW and operating
between 120 and 15,000 rpm. Within ISO 10816-3, criteria are
established for four different groups of machinery, and
provisions include either flexible or rigid support conditions.
Criteria are also established based on both rms velocity and rms
displacement. Part 4, ISO 10816-4, identifies evaluation criteria
for gas-turbine-driven power generation units (excluding
aircraft derivatives) operating between 3000 and 20,000 rpm.
Part 5 (ISO 10816-5) applies to machine sets in hydro-power
facilities and pumping plants. Part 6, ISO 10816-6, provides
evaluation criteria for reciprocating machines with power
ratings over 134 horsepower (100 kW). The scope of Part 5 is
not applicable to general equipment foundations and the
criteria of Part 6 are not sufficiently substantiated and defined
to be currently useful.
Another document available for establishing vibration
limitation is from Lifshits (Lifshits, Simmons, and Smalley
1986). This document follows Blake’s approach of identifying five different categories from No Faults to Danger of
Immediate Failure. In addition, a series of correction factors
are established to broaden the applicability to a wider range
of equipment and measurement data.
Blake’s paper (Blake 1964) has become a common basis
for some industries and firms. His work presented a standard
vibration chart for process equipment with performance
rated from “No Faults (typical of new equipment)” to
“Dangerous (shut it down now to avoid danger).” The chart
was primarily intended to aid plant personnel in assessing field
installations and determining maintenance plans. Service
factors for different types of equipment are used to allow
widespread use of the basic chart. This tool uses vibration
displacement (in. or mm) rather than velocity and covers speed
ranges from 100 to 10,000 rpm. Figure 3.9 and Table 3.4
present the basic chart and service factors established by Blake.
Baxter and Bernhard (1967) offered more general vibration
tolerances in a paper that has also become widely referenced.
Again with primary interest to the plant maintenance operations,
they established the General Machinery Vibration Severity
Chart, shown in Fig. 3.10, with severity ranging from extremely
FOUNDATIONS FOR DYNAMIC EQUIPMENT
smooth to very rough. These are plotted as displacement versus
vibration frequency so that the various categories are differentiated along lines of constant peak velocity.
The American Petroleum Institute (API) also has a series
of standards for equipment common in the petrochemical
industry (541, 610, 612, 613, 617, 618, and 619). ISO 10816-3
can be applied for some large electrical motors; however,
most design offices do not generally perform rigorous analyses
for these items.
Figure 3.11 provides a comparison of five generic standards
against four corporate standards. To the extent possible, the
comparisons are presented on a common basis. In particular,
the comparison is based on equipment that is in service,
perhaps with minor faults, but which could continue in
service indefinitely. The Blake line is at the upper limit of the
zone identifying operation with minor faults with a service
factor of one applicable for fans, some pumps, and similar
equipment. The Lifshits line separates the acceptable and
marginal zones and includes a K of 0.7, reflecting equipment
with rigid rotors. The ISO lines are drawn at the upper level
of Zone B, normally considered acceptable for long-term
operation. The ISO 10816-3 line is for large machines
between 400 and 67,000 horsepower (300 kW and 50 MW)
on rigid support systems. The ISO 10816-2 is for large
turbines over 67,000 horsepower (50 MW).
The company standards are used for comparison to calculate
motions at the design stage. For these calculations, the companies prescribe rotor unbalance conditions worse than those
expected during delivery and installation. These load definitions are consistent with those presented in Section 3.2.2.1.
Thus, there is a level of commonality. Company G’s criteria
are for large turbine applications and, thus, most comparable
to the ISO 10816-2 criteria. The other company standards are
for general rotating equipment. Company F permits higher
motions for reciprocating equipment. In all cases the design
companies standards reflect that the manufacturer may
establish equipment-specific criteria that could be more
limiting than their internal criteria.
Figure 3.11 shows that the corporate standards are generally
below the generic standards because the generic standards are
intended for in-place service checks and maintenance decisions
rather than offering initial design criteria. One company is
clearly more lenient for very low-speed equipment, but the
corporate standards tend to be similar.
The Shock and Vibration Handbook (Harris 1996)
contains further general information on such standards.
3.4.2 Physiological limits—Human perception and sensitivity to vibration is ambiguous and subjective. Researchers
have studied and investigated this topic, but there are no clear
uniform U.S. standards. In Germany, VDI 2057 provides
guidance for the engineer. Important issues are the personnel
expectations and needs and the surrounding environment.
ISO 2631 provides guidance for human exposure to
whole-body vibration and considers different comfort levels
and duration of exposure. This document does not address
the extensive complexities identified in ISO 2631. Figure 3.12
presents the basic suggested acceleration limits from ISO
2631 applicable to longitudinal vibrations (vertical for a
351.3R-21
Fig. 3.10—General Machinery Vibration Severity Chart
(Baxter and Bernhard 1967).
Fig. 3.11—Comparison of permissible displacements.
standing person). This figure reflects the time of exposure and
frequency consideration for fatigue-decreased proficiency.
The figure shows that people exhibit fatigue and reduced proficiency when subjected to small accelerations for long periods or
greater accelerations for shorter periods. The frequency of the
accelerations also impact fatigue and proficiency.
The modified Reiher-Meister figure (barely perceptible,
noticeable, and troublesome) is also used to establish limits
with respect to personnel sensitivity, shown in Fig. 3.13.
351.3R-22
ACI COMMITTEE REPORT
Table 3.5—Short-term permissible values
(DIN 4150-3)
Type
Foundation Foundation
of building (1 to 10 Hz) (10 to 50 Hz)
Top complete
Foundation
floor (all
(50 to 100 Hz) frequencies)
Industrial and 0.8 in./s
0.8 to 1.6 in./s 1.6 to 2.0 in.s
commercial (20 mm/s) (20 to 40 mm/s) (40 to 50 mm/s)
Residential
Special or
sensitive
Fig. 3.12—Longitudinal acceleration limits (adapted from
ISO 2631-1).
Fig. 3.13—Reiher-Meister Chart (Richart, Hall, and
Woods 1970).
DIN 4150 is another standard used internationally. Part 3
defines permissible velocities suitable for assessment of shortterm vibrations on structures, which are given in Table 3.5.
Furthermore, Part 2 of this German standard defines limitations
for allowable vibrations based on perception as a function of
location (residential, light industrial) and either daytime or
nighttime. Most engineering offices do not consider human
perception to vibrations, unless there are extenuating circumstances (proximity to office or residential areas).
There are no conclusive limitations on the effects of vibration
of surrounding buildings. The Reiher-Meister figure identifies
0.2 in./s
(5 mm/s)
0.1 in./s
(3 mm/s)
0.2 to 0.6 in./s
(5 to 15 mm/s)
0.1 to 0.3 in./s
(3 to mm/s)
0.6 to 0.8 in./s
(15 to 20 mm/s)
0.3 to 0.4 in./s
(8 to 10 mm/s)
1.6 in./s
(40 mm/s)
0.6 in./s
(15 mm/s)
0.3 in./s
(8 mm/s)
levels of vibration from mining operations that have
damaged structures.
3.4.3 Frequency ratios—The frequency ratio is a term that
relates the operating speed of the equipment to the natural
frequencies of the foundation. Engineers or manufacturers
require that the frequency of the foundation differ from the
operating speed of the equipment by certain margins. This
limitation is applied to prevent resonance conditions from
developing within the dynamic soil-foundation-equipment.
The formulation or presentation of frequency ratios may be
based around either fo/fn or fn/fo (operating frequency to
natural frequency or its inverse), and engineers or manufacturers should exercise caution to prevent misunderstandings.
A common practice among engineering firms is to
compute the natural frequencies of the basic equipmentfoundation and compare the values with the dynamic excitation
frequency. Many companies require that the natural
frequency be 20 to 33% removed from the operating speed.
Some firms have used factors as low as 10% or as high as
50%. If the frequencies are well separated, no further evaluation
is needed. If there is a potential for resonance, the engineer
should either adjust to the foundation size or perform more
refined calculations. Refined calculations may include an
analysis with a deliberately reduced level of damping. The
size and type of equipment play an important role in this
decision process.
Frequency ratio is a reasonable design criterion, but one
single limiting value does not fit all situations. Where there
is greater uncertainty in other design parameters (soil stiffness,
for example), more conservatism in the frequency ratio may
be appropriate. Similarly, vibration problems can exist even
though resonance is not a problem.
3.4.4 Transmissibility—A common tool for the assessment
of vibrations at the design stage is a transmissibility ratio, as
shown in Fig. 3.14, which is based on a single degree-offreedom (SDOF) system with a constant speed fo excitation
force. This ratio identifies the force transmitted through the
spring-damper system with the supporting system as
compared with the dynamic force generated by the equipment. This ratio should be as low as possible, that is, only
transmit 20% of the equipment dynamic force into the
supporting system. Low transmissibility implies low vibrations in the surroundings, but this is not an absolute truth.
This transmissibility figure assumes that the damping force is
directly and linearly proportional to the velocity of the SDOF.
Where the system characteristics are such that the damping
force is frequency dependent, the aforementioned represen-
FOUNDATIONS FOR DYNAMIC EQUIPMENT
351.3R-23
tation is not accurate. When the damping resistance
decreases at higher frequencies, the deleterious effect of
damping on force transmissibility can be mitigated.
For soil or pile-supported systems, the transmissibility
ratio may not be meaningful. In SDOF models of these
systems, the spring and damper are provided by the soil and,
while the transmissibility of the design may be low, the
energy worked through these system components is motion
in the surroundings that may not be acceptable.
3.5—Concrete performance criteria
The design of the foundation should withstand all applied
loads, both static and dynamic. The foundation should act in
unison with the equipment and supporting soil or structure to
meet the deflection limits specified by the machinery manufacturer or equipment owner.
The service life of a concrete foundation should meet or
exceed the anticipated service life of the equipment installed
and resist the cyclic stresses from dynamic loads. Cracking
should be minimized to ensure protection of reinforcing steel.
The structural design of all reinforced concrete foundations
should be in accordance with ACI 318. The engineer may use
allowable stress methods for nonprestressed reinforced concrete.
In foundations thicker than 4 ft (1.2 m), the engineer may
use the minimum reinforcing steel suggested in ACI 207.2R.
API and the Construction Industry Institute published API
Recommended Practice 686/PIP REIE 686, “Recommended
Practice for Machinery Installation and Installation Design.”
Chapter 4 of 686/PIP REIE 686 includes design criteria for
soil-supported reinforced concrete foundations that supports
general and special purpose machinery. The concrete used in
the foundation should tolerate its environment during placement, curing, and service. The engineer should consider
various exposures such as freezing and thawing, salts of
chlorides and sulfates, sulfate soils, acids, carbonation,
repeated wetting and drying, oils, and high temperatures.
In addition to conventional concrete, there are many technologies available—such as admixtures, additives, specialty
cements, and preblended products—to help improve placement, durability, and performance properties. These additives include water reducers, set-controlling mixtures,
shrinkage-compensating admixtures, polymers, silica fumes,
fly ash, and fibers.
Many foundations, whether new or repaired, require a fast
turnaround to increase production by reducing downtime
without compromising durability and required strength.
These systems may use a combination of preblended or
field-mixed concrete and polymer concrete or grout to
reduce downtime to 12 to 72 h, depending on foundation
volume and start-up strength requirements.
3.6—Performance criteria for machinemounting systems
The machine-mounting system (broadly categorized as
either an anchorage-type or an isolator-type) attaches the
dynamic machine to its foundation. It represents a vital
interface between the machine and the foundation; however,
it can suffer from insufficient attention to critical detail by
Fig. 3.14—Force transmissibility.
the foundation engineer and machinery engineer because it
falls between their areas of responsibility. Anchorage-type
machine-mounting systems integrate the foundation and the
machine into a single structure. Isolator-type machinemounting systems separate the machine and the foundation
into two separate systems that may still dynamically interact
with each other. In the processes of design, installation, and
operation, the critical role of both types needs advocacy and
the assurance that interface issues receive attention. The
research and development of information on machinemounting system technology by the Gas Machinery
Research Council (GMRC) during the 1990s reflects the
importance that this group attaches to the anchorage-type
machine-mounting system. This research produced a series
of reports on machine-mounting topics (Pantermuehl and
Smalley 1997a,b; Smalley and Pantermuehl 1997; Smalley
1997). These reports, readily retrievable from www.gmrc.org,
are essential for those responsible for dynamic machines and
their foundations.
Most large machines, in spite of careful design for integrity
and function by their manufacturers, can internally absorb no
more than a fraction of the forces or thermal growth inherent
in their function.
Those responsible for the machine-foundation interface
should provide an attachment that transmits the remaining
forces for dynamic integrity of the structure yet accommodates anticipated differential thermal expansion between
machine and foundation. They should recognize the inherent
conflicts in these requirements, the physical processes that
can inhibit performance of these functions, and the lifetime
constraints (such as limited maintenance and contaminating
materials) from which any dynamic machine can suffer as it
contributes to profitable, productive plant operation.
A dynamic machine may tend to get hotter and grow more
than its foundation (in the horizontal plane). The growth can
reach several tenths of an inch (0.1 in. = 2.54 mm); combustion
turbine casings grow so much that they have to include deliberately installed flexibility between hotter and cooler elements
of their own metallic structure. Most machines—such as
compressors, steam turbines, motors, and generators—do
not internally relieve their own thermal growth, so the
mounting system should allow for thermal growth. Thermal
growth can exert millions of pounds of force (1,000,000 lbf
= 4500 kN), a level that cannot be effectively restrained.
351.3R-24
ACI COMMITTEE REPORT
Heat is transferred between the machine and foundation
through convection, radiation, and conduction. While
convection and radiation dominate in the regions where an
air gap separates the machine base from the foundation, the
mounting system provides the primary path for conduction.
Ten critical performance criteria can be identified as
generally applicable to isolator and anchorage-type
mounting systems:
1. A machine-mounting system should tolerate expected
differential thermal growth across the interface. This can
occur by combining strength to resist expansion forces and
stresses, flexibility to accommodate the deflections, and
tolerance for relative sliding across the interface (as the
machine grows relative to the foundation);
2. A machine-mounting system should either absorb or
transmit, across the mounting interface, those internally
generated dynamic forces, resulting from the machine’s
operation not absorbed within the machine structure itself.
These forces include both vertical and horizontal components. Flexible mounts that deflect rather than restrain the
forces become an option only in cases where the machine
and any rigidly attached structure have the structural rigidity
needed to avoid damaging internal stresses and deflections.
Large machinery may not meet this criterion;
3. A nominally rigid mount should transmit dynamic
forces with only microlevel elastic deformation and negligible
dynamic slippage across the interface. The dynamic forces
should include local forces, such as forces from each individual
cylinder, which large machines transmit to the foundation
because of their flexibility. For reciprocating compressors,
this criterion helps ensure that the foundation and machine
form an integrated structure;
4. A machine-mounting system should perform its function
for a long life—typically 25 years or more. Specifications
from the operator should include required life;
5. Any maintenance and inspection required to sustain
integrity of the machine-mounting system should have a
frequency acceptable to the operator of the machine, for
example, once per year. Engineers, installers, and operators of
the machine and its foundation should agree to this maintenance requirement because the design integrity relies on the
execution of these maintenance functions with this frequency;
6. The bolts that tie the machine to the mounting system,
and which form an integral part of the mounting system,
should have sufficient stretch and create enough normal
force across all interfaces to meet the force transmission and
deflection performance stated above;
7. The anchor bolt material strength should tolerate the
resultant bolt tensile stresses. The mounts, soleplates, and
grout layers compressed by the anchor bolt should tolerate
the compressive stresses imposed on them;
8. Any polymeric material (grout or chocks) compressed
by the anchor bolts should exhibit a tolerably low amount of
creep to maintain bolt stretch over the time period between
maintenance actions performed to inspect and tighten anchor
bolts. Indeed, the machine mount should perform its function, accounting for expected creep, even if maintenance
occurs less frequently;
9. The mounting system should provide a stable platform
from which to align the machine. Any deflections of the
mounting system that occur should remain sufficiently
uniform at different points to preserve acceptable alignment
of the machine. The specifications and use of adjustable
chock mounts has become increasingly widespread to
compensate for loss of alignment resulting from creep and
other permanent deformations; and
10. The mounting system should impose tolerable loads,
stresses, and deformations on the foundation itself. Appropriate foundation design to make the loads, stresses, and
deformations tolerable remains an essential part of this
performance criterion. Some of the loads and stresses to
consider include:
• Tensile stresses in the concrete at the anchor bolt termination point, which may cause cracks;
• Shear stresses in concrete above anchor bolt termination
points, which, if high enough, might result in pullout;
• Interface shear stresses between a grout layer and the
concrete resulting from the typically higher expansion
of polymer grout than concrete (best accommodated
with expansion joints); and
• Hogging or sagging deformation of the concrete block
produced by heat conduction through the mounting
system. Air gaps and low conductivity epoxy chocks
help minimize such deformation.
Potential conflicts requiring attention and management in
these performance criteria include:
• Requirements to accommodate thermal expansion
while transmitting dynamic forces; and
• Requirements to provide a large anchor bolt clamping
force (so that slippage is controlled during transfer of
high lateral loads) while stresses and deflections in bolt,
foundation, chocks, and grout remain acceptably low.
Physical processes that can influence the ability of the
mount to meet its performance criteria include:
• Creep—Creep of all polymeric materials under compressive load. (Creep means time-dependent deflection
under load. Deflection increases with time—sometimes
doubling or tripling the initial deflection);
• Differential thermal expansion—This can occur when
two adjacent components at similar temperatures have
different coefficients of thermal expansion, when two
adjacent components of similar coefficients have different
temperatures or a combination of both. Machine
mounts with epoxy materials can experience both types
of differential thermal expansion;
• Friction—Friction is limited by a friction coefficient.
Friction defines the maximum force parallel to an
interface that the interface can resist before sliding for a
given normal force between the two interfacing materials;
• Limits on friction—The presence of oil in the interface
(typically cutting the dry friction coefficient in half)
causes further limits on friction;
• Yield strength—Yield strength of anchor bolts limits the
tension available from an anchor bolt and encourages
the use of high-strength anchor bolts for all critical
applications;
FOUNDATIONS FOR DYNAMIC EQUIPMENT
•
Cracks—Concrete can crack under tensile loads, and
these cracks can grow with time; and
• Oil—Oil can pool around many machinery installations.
Oil aggravates cracks in concrete, particularly under
alternating stresses where it induces a hydraulic action. In
many cases, oil, its additives, or the ambient materials it
transports react with concrete to reduce its strength, particularly in cracks where stresses tend to be high.
Those responsible for machine mounts, as part of a foundation, should consider the aforementioned performance criteria,
the conflicts that complicate the process of meeting those
criteria, and the physical processes that inhibit the ability of any
installation to meet the performance criteria. Other sections of
this document address the calculation of loads, stresses, deflections, and the specific limits of strength implicit in different
materials. The GMRC reports referred to address all these
issues as they pertain to reciprocating compressors.
2
(3-38)
2
ri ωo
Fzi
′′ = (mrot,i )ri ---------- cos2(ωot + αi)
Li
(3-42)
If the i-th cylinder is oriented at angle θi to a global horizontal
z-axis, then the primary and secondary force components,
with respect to the global axis, can be rewritten as follows
(primary)
GP
(3-43)
GP
(3-44)
F zi = F′zi cos θ i = F′yi sin θ i
F yi = F′zi sin θ i = F′yi cos θ i
(secondary)
GS
(3-45)
GS
(3-46)
F zi = F′′
zi cos θ i
3.7—Method for estimating inertia forces
from multicylinder machines
The local horizontal Fzi and vertical Fyi unbalanced forces
for the i-th cylinder located in the horizontal plane can be
written as
Fzi = (mrec,i + mrot,i )riωo2 cos(ωot + αi)
351.3R-25
F yi = F′′
zi sin θ i
The resultant forces due to n cylinders in global coordinates
can be calculated as follows
n
2
GP
2
ri ωo
- cos2(ωot + αi )
+ mrot,i ---------Li
Fz
=
∑ Fzi
GP
(3-47)
i=1
n
and
GS
Fz
2
F yi = m rec, i r i ω o sin ( ω o t + α i )
=
∑ Fzi
GS
(3-48)
i=1
(3-39)
n
GP
Fy
where
mrec,i =
mrot,i =
=
ri
Li
=
reciprocating mass for the i-th cylinder;
rotating mass of the i-th cylinder;
radius of the crank mechanism of the i-th cylinder;
length of the connecting rod of the crank mechanism at the i-th cylinder;
ωo = circular operating frequency of the machine (rad/s);
t
= time, s; and
αi
= the phase angle for the crank radius of the i-th
cylinder, rad.
The primary and secondary force components are as
follows
(primary)
=
∑ Fyi
GP
(3-49)
i=1
n
GS
Fy
=
∑ Fyi
GS
(3-50)
i=1
The resultant moments due to n cylinders in global coordinates can be determined as follows
n
GP
My
=
∑ ( Fzi
GP
X i ) (moment about y [vertical] axis) (3-51)
i=1
n
2
Fzi
′ = (mrec,i + mrot,i )ri ω o cos(ωot + αi)
GS
My
(3-40)
=
∑ ( Fzi Xi )
GS
(3-52)
i=1
2
′ = mrec,i ri ω o sin(ωot + αi)
Fyi
(3-41)
n
GP
Mz
(secondary)
=
∑ ( Fyi
GP
i=1
X i )(moment about z [horizontal] axis) (3-53)
351.3R-26
ACI COMMITTEE REPORT
GS
n
GS
Mz
=
∑
GS
( F yi X i )
i=1
Maximum global vertical primary moment:
where Xi = distance along the crankshaft from the reference
origin to the i-th cylinder.
Equation (3-43) to (3-54) provide instantaneous values of
time-varying inertia (shaking) forces and four time varying
shaking moments for an n cylinder reciprocating machine.
To visualize the time variation of these forces and moments
over a revolution of the crankshaft, they can be computed at
a series of crank angle values and plotted against crank
angle. To obtain maximum values of the primary and
secondary forces and moments (and the phase angle at which
the maxima occur), they are computed at two orthogonal
angles and vectorially combined as shown as follows
Maximum global horizontal primary force:
GP
GP 2
GP 2 1 ⁄ 2
( F Z ) max = [ ( F Z0 ) + ( F Z90 ) ]
GP
(3-55)
GP
at tan–1 ( F Z90 ⁄ F Z0 )
Maximum global horizontal secondary force:
GS
GS 2 1 ⁄ 2
GS 2
( F Z ) max = [ ( F Z0 ) + ( F Z45 ) ]
GS
(3-56)
GS
at tan–1 ( F Z45 ⁄ F Z0 )
Maximum global vertical primary force:
GP
GP 2
GP 2 1 ⁄ 2
( F Y ) max = [ ( F Y0 ) + ( F Y90 ) ]
GP
(3-57)
GP
at tan–1 ( F Y90 ⁄ F Y0 )
Maximum global vertical secondary force:
GS
GS 2 1 ⁄ 2
GS 2
( F Y ) max = [ ( F Y0 ) + ( F Y45 ) ]
GS
(3-58)
GS
at tan–1 ( F Y45 ⁄ F Y0 )
Maximum global horizontal primary moment:
GP
GP 2
GP 2 1 ⁄ 2
( M Z ) max = [ ( M Z0 ) + ( M Z90 ) ]
GP
(3-59)
GP
at tan–1 ( M Z90 ⁄ M Z0 )
Maximum global horizontal secondary moment:
GS
GS
at tan–1 ( M Z45 ⁄ M Z0 )
(3-54)
GS 2
GS 2 1 ⁄ 2
( M Z ) max = [ ( M Z0 ) + ( M Z45 ) ]
(3-60)
GP
GP 2
GP 2 1 ⁄ 2
( M Y ) max = [ ( M Y0 ) + ( M Y90 ) ]
GP
(3-61)
GP
at tan–1 ( M Y90 ⁄ M Z0 )
Maximum global vertical secondary moment:
GS
GS 2 1 ⁄ 2
GS 2
( M Y ) max = [ ( M Y0 ) + ( M Y45 ) ]
GS
(3-62)
GS
at tan–1 ( M Y45 ⁄ F Y0 )
where subscripts 0, 45, and 90 represent the value of ωot used to
calculate the force values listed in Eq. (3-55) to (3-62).
CHAPTER 4—DESIGN METHODS AND MATERIALS
4.1—Overview of design methods
4.1.1 General considerations—The objectives of the
machine foundation design are to assess the dynamic
response of the foundation and verify compliance with the
required vibration and structural performance criteria.
Machine foundation design includes the following steps:
1. Develop a preliminary size for the foundation using
rule-of-thumb approaches, past experience, machine manufacturer recommendations, and other available data;
2. Calculate the vibration parameters, such as natural
frequency, amplitudes, velocities, and accelerations, for the
preliminarily sized foundation;
3. Verify that these calculated parameters do not exceed
recommended limits or vibration performance criteria;
4. If necessary, incorporate appropriate modifications in
the foundation design to reduce vibration responses to meet
the specified vibration performance criteria and cost; and
5. Check the structural integrity of the concrete foundation
and machine-mounting system.
Preventive measures to reduce vibrations are less expensive
when incorporated in the original design than remedial
measures applied after machinery is in operation. The
following are some common means that can be used separately
or combined to reduce vibrations:
• Selection of the most favorable location for the machinery;
• Adjustment of machine with respect to speed or balance
of moving parts;
• Adjustment of the foundation with respect to mass
(larger versus smaller) or foundation type (soil supported
versus pile);
• Isolation of the machinery from the foundation using
special mountings such as springs and flexible mats; and
• Isolation of the foundations by barriers.
4.1.2 Summary of design methods for resisting dynamic
loads—Design methods for the foundations supporting
dynamic equipment have gradually evolved over time from
FOUNDATIONS FOR DYNAMIC EQUIPMENT
an approximate rule-of-thumb procedure to the scientifically
sound engineering methods. These methods can be identified
as follows:
• Rule-of-thumb;
• Equivalent static loading; and
• Dynamic analysis.
The selection of an appropriate method depends heavily
on machine characteristics, including unbalanced forces,
speed, weight, center of gravity location, and mounting;
importance of the machine; foundation type and size; and
required performance criteria.
Design of the foundation starts with the selection and
assessment of the foundation type, size, and location. Usually,
foundation type is governed by the soil properties and
operational requirements. The machine footprint, weight, and
unbalanced forces govern the size of the foundation. The
location of a foundation is governed by environmental and
operational considerations. Thus, the engineer should consider
information from the following three categories before the
foundation can be preliminarily sized:
1. Machine characteristics and machine-foundation
performance requirements
• Functions of the machine;
• Weight of the machine and its moving components;
• Location of the center of gravity in both vertical and
horizontal dimensions;
• Speed ranges of the machine;
• Magnitude and direction of the unbalanced forces
and moments; and
• Limits imposed on the foundation with respect to
differential displacement.
2. Geotechnical information
• Allowable soil-bearing capacity;
• Effect of vibration on the soil, for example, settlement or liquefaction risk;
• Classification of soil;
• Modulus of subgrade reaction;
• Dynamic soil shear modulus; and
• Dynamic soil-pile interaction parameters (for pilesupported foundations).
3. Environmental conditions
• Existing vibrating sources such as existing vibration
equipment, vehicular traffic, or construction;
• Human susceptibility to vibration or vibration-sensitive equipment;
• Flooding or high water table risk; and
• Seismic risk.
Usually, the preliminary foundation size is established
using the rule-of-thumb method, and then the performance
criteria, for both machine and foundation, are verified using
the equivalent static loading method or dynamic analysis. If
the equivalent static loading method or dynamic analysis
shows that the foundation is inadequate, the engineer revises
the foundation size and repeats the analysis.
4.1.2.1 Rule-of-thumb method—Rule-of-thumb is one of
the simplest design methods for machine foundations resisting
vibrations. The concept of this method is to provide sufficient
351.3R-27
mass in the foundation block so that the vibration waves are
attenuated and absorbed by the block and soil system.
Most engineers consider the rule-of-thumb procedure
satisfactory for a preliminary foundation sizing. Sometimes
engineers use this method to design block-type foundations
supporting relatively small machinery, up to 5000 lbf (22 kN)
in weight and having small unbalanced forces. For reciprocating machinery and sensitive machinery, rule-of-thumb
procedures by themselves may not be sufficient.
A long-established rule-of-thumb for machinery on blocktype foundations is to make the weight of the foundation
block at least three times the weight of a rotating machine
and at least five times the weight of a reciprocating machine.
For pile-supported foundations, these ratios are sometimes
reduced so that the foundation block weight, including pile
cap, is at least 2-1/2 times the weight of a rotating machine
and at least four times the weight of a reciprocating machine.
These ratios are machine weights inclusive of moving and
stationary parts as compared with the weight of the concrete
foundation block. Additionally, many designers require the
foundation to be of such weight that the resultant of lateral
and vertical loads falls within the middle third of the foundation
base. That is, the net effect of lateral and vertical loads or the
eccentricity of the vertical load should not cause uplift.
The engineer should size the shape and thickness of the
foundation to provide uniform distribution of vertical dead
and live loads to the supporting soil or piles, if practical. The
shape of the foundation should fit the supported equipment
requirements. Also, the engineer should provide sufficient
area for machine maintenance. The shape of the foundation
should adequately accommodate the equipment, including
maintenance space if required. Minimum width should be
1.5 times the vertical distance from the machine centerline to
the bottom of the foundation block. The designer should
adjust the length and width of the foundation so the center of
gravity of the machine coincides with the center of gravity of
the foundation block in plan. A common criteria is that the
plan view eccentricities between the center of gravity of the
combined machine-foundation system and the center of
resistance (center of stiffness) should be less than 5% of the
plan dimensions of the foundation. In any case, the foundation
is sized so that the foundation bearing pressure does not
exceed the allowable soil-bearing capacity.
Thickness criteria primarily serve to support a common
assumption that the foundation behaves as a rigid body on
the supporting material. Clearly, this is a more complex
problem than is addressed by simple rules-of-thumb. On soft
materials, a thinner section may be sufficient, whereas on
stiffer soils, a thicker section might be required to support the
rigid body assumption. If the rigid body assumption is not
applicable, more elaborate computation techniques, such as
finite element methods, are used. Gazetas (1983) provides
some direction in this regard. One rule-of thumb criterion for
thickness is that the minimum thickness of the foundation block
should be 1/5 of its width (short side), 1/10 of its length (long
side), or 2 ft (0.6 m), whichever is greatest. Another criterion is
given in Section 4.3 as 1/30 of the length plus 2 ft (0.6 m).
351.3R-28
ACI COMMITTEE REPORT
The designer may need to provide isolation or separation
of the machine foundation from the building foundation or
slab. Separation in the vertical direction may also be appropriate. Normally, dynamically loaded foundations are not
placed above building footings or in such locations that the
dynamic effects can transfer into the building footings.
4.1.2.2 Equivalent static loading method—The equivalent
static loading method is a simplified and approximate way of
applying pseudodynamic forces to the machine-supporting
structure to check the strength and stability of the foundation.
This method is used mainly for the design of foundations for
machines weighing 10,000 lbf (45 kN) or less.
For design of reciprocating machine foundations by the
static method, the machine manufacturer should provide the
following data:
• Weight of the machine;
• Unbalanced forces and moments of the machine during
operation; and
• Individual cylinder forces including fluid and inertia
effects.
For design of rotating machine foundations by the static
method, the machine manufacturer should provide:
• Weight of the machine and base plate;
• Vertical pseudodynamic design force*; and
• Horizontal pseudodynamic design forces†—lateral
force and longitudinal force.
Calculated natural frequencies, deformations, and forces
within the structure supporting the machine should satisfy
established design requirements and performance criteria
outlined in Section 3.4.
4.1.2.3 Dynamic analysis—Dynamic analysis incorporates
more advanced and more accurate methods of determining
vibration parameters and, therefore, is often used in the final
design stage and for critical machine foundations. Dynamic
analysis (or vibration analysis) is almost always required for
large machine foundations with significant dynamic forces
(Section 4.3).
For the dynamic analysis of reciprocating and rotating
machine foundations, the machine manufacturer should provide
the following data to the extent required by the analysis:
• Primary unbalanced forces and moments applied at the
machine speed over the full range of specified operating
speeds;
• Secondary unbalanced forces and moments applied at
twice the machine speed over the full range of specified
operating speeds; and
• Individual cylinder forces including fluid and inertia
effects.
*Vertical dynamic force is applied normal to the shaft (for a horizontally shafted
machine) at the midpoint between the bearings or at the centroid of the rotating mass.
If the magnitude of this force is not provided by the machine manufacturer, the engineer
may conservatively take it as 50% of the machine assembly dead weight.
†
Lateral force is applied normal to the shaft at the midpoint between the bearings or
at the centroid of the rotating mass. Longitudinal force is applied along the shaft axis.
The engineer can conservatively take both the lateral and the longitudinal force as
25% of the dead weight of the machine assembly, unless specifically defined by the
machine manufacturer. Vertical, lateral, and longitudinal forces are not considered to
act concurrently for many types of rotating machines.
For the rotating machinery, the dynamic design (Section 4.3)
determines the vibration amplitudes based on these dynamic
forces. Refer to Section 3.2 for the dynamic force calculations
for both reciprocating and rotating machines. When there is
more than one rotor, however, amplitudes are often
computed with the rotor forces assumed to be in-phase and
180 degrees out-of-phase. To obtain the maximum translational
and maximum torsional amplitudes, other phase relationships
may also be investigated.
A complete dynamic analysis of a system is normally
performed in two stages:
1. Determination of the natural frequencies and mode
shapes of a machine-foundation system; and
2. Calculation of the machine-foundation system response
caused by the dynamic forces.
Determining the natural frequencies and mode shapes of the
machine-foundation system provides information about the
dynamic characteristic of that system. In addition, calculating
the natural frequencies identifies the fundamental frequency,
usually the lowest value of the natural frequencies.
The natural frequencies are significant for the following
reasons:
• They indicate the relative degree of stiffness of the
machine-foundation-soil system; and
• They can be compared with the frequency of the acting
dynamic force so that a possible resonance condition
may be prevented. Resonance is prevented when the
ratio of the machine operating frequency to the fundamental frequency of the machine-foundation system
falls outside the undesirable range (Section 3.4.3). In
many cases, six frequencies are calculated consistent
with six rigid body motions of the overall machinefoundation system. Any or all of the six frequencies
may be compared with the excitation frequency when
checking for resonance conditions.
The significance of the mode shapes is as follows:
• They indicate the deflection pattern that the machinefoundation system assumes when it is left to vibrate
after the termination of the disturbing force. Generally,
it is the first mode that dominates the vibrating shape,
and the higher modes supplement that shape when
superimposed;
• They indicate the relative degree of the structural stiffness
among various points of the machine-foundation system,
that is, the relationship between different amplitudes
and mode shapes. If the flexural characteristics of a
foundation are being modeled, a mode shape may indicate
that a portion of the foundation (a beam or a pedestal) is
relatively flexible at a particular frequency. For a system
represented by a six degree-of-freedom model, the relative
importance of the rocking stiffness and the translational
stiffness can be indicated;
• They can be used as indicators of sensitivity when varying
the stiffness, mass, and damping resistance of the
machine-foundation system to reduce the vibration amplitudes at critical points. A particular beam or foundation
component may or may not be significant to the overall
behavior of the system. An understanding of the mode
FOUNDATIONS FOR DYNAMIC EQUIPMENT
351.3R-29
shape can aid in identifying the sensitive components.
Calculation of the machine-foundation system response
caused by the dynamic forces provides the vibration parameters—such as displacement, velocity, and acceleration of
the masses—and also the internal forces in the members of
the machine support system. Then, these vibration parameters are compared with the defined criteria or recommended
allowable values for a specific condition (Sections 4.3 and 3.4),
and internal forces are used to check the structural strength
of the foundation components.
4.2—Impedance provided by the supporting media
The dynamic response of foundations depends on stiffness
and damping characteristics of the machine-foundation-soil
system. This section presents a general introduction to this
subject and a summary of approaches and formulae often
used to evaluate the stiffness and damping of both soilsupported and pile foundations. These stiffness and damping
relationships, collectively known as impedance, are used for
determining both free-vibration performance and motions of
the foundation system due to the dynamic loading associated
with the machine operation.
The simplest mathematical model used in dynamic analysis
of machine-foundation systems is a single degree-offreedom representation of a rigid mass vertically supported
on a single spring and damper combination (Fig. 4.1(a)).
This model is applicable if the center of gravity of the
machine-foundation system is directly over the center of
resistance and the resultant of the dynamic forces (acting
through a center of force, CF) is a vertical force passing
through center of gravity. The vertical impedance (kv*) of
the supporting media is necessary for this model.
The next level of complexity is a two degree-of-freedom
model commonly used when lateral dynamic forces act on a
machine-foundation system (Fig. 4.1(b)). Because the lines
of action of the forces and the resistance do not coincide, the
rocking and translational motions of the system are coupled.
For this model, the engineer needs to calculate the horizontal
translational impedance (ku*) and the rocking impedance
(kψ*) of the supporting media. These impedance values,
especially the rocking terms, are usually different in
different horizontal directions.
Application of the lateral dynamic forces can cause a
machine-foundation system to twist about a vertical axis. To
model this behavior, the engineer needs to determine the
torsional impedance (kη*) of the supporting media. As in the
vertical model, if the in-plan eccentricities between the center
of gravity and center of resistance are small, this analysis is
addressed with a single degree-of-freedom representation.
When the machine-foundation system is rigid and on the
surface, the impedance (ku1*, ku2*, kv*, kψ1*, kψ2*, kη*) can
be represented as six elastic springs (three translational and
three angular) and six viscous or hysteretic dampers (three
translational and three angular), which act at the centroid of
the contact area between the foundation and the supporting
media (Fig. 4.1(c)). If the support is provided by piles or
isolators, the equivalent springs and dampers act at the center
of stiffness of those elements. When the foundation is
Fig. 4.1—Common dynamic models.
embedded, additional terms can be introduced as spring and
damper elements acting on the sides of the foundation. The
effective location of the embedment impedance is determined
rationally based on the character of the embedment (Beredugo
and Novak 1972; Novak and Beredugo 1972; Novak and Sachs
1973; Novak and Sheta 1980; Lakshmanan and Minai 1981).
The next level of complexity addresses flexible foundations.
The structural representation of these foundations is typically
modeled using finite element techniques. The flexibility of the
structure might be represented with beam-type elements, twodimensional plane stress elements, plate bending elements,
three-dimensional elements, or some combination of these.
The supporting media for these flexible foundations is
addressed in one of three basic manners. The most simplistic
approach uses a series of simple, elastic, translational springs
at each modeled contact point between the structure and the
soil. These spring parameters are computed from the spring
constants determined assuming a rigid foundation assumption
or by some similar rational procedure. The other two
approaches are finite element modeling and modeling with
boundary elements. Such approaches are not commonly used
in day-to-day engineering practice but can be used successfully for major, complex situations where a high degree of
refinement is justified.
The basic mathematical model used in the dynamic analysis
of various machine-foundation systems is a lumped mass
with a spring and dashpot (Fig. 4.2). Lumped mass includes
the machine mass, foundation mass, and, in some models,
soil mass. A mass m, free to move only in one direction, for
351.3R-30
ACI COMMITTEE REPORT
below the mass is kstv for any displacement v. Nevertheless,
the total restoring force generated by the column at the top of
the lumped mass is the sum of the elastic force in the column
and the inertia force of the mass. If the displacement of this
mass varies harmonically
v(t) = v ⋅ cosωmt
(4-1)
where
v
= displacement amplitude; and
ωm = circular frequency of motion.
Then, the acceleration is
2
2
d v
v·· = -------- = – v ⋅ ω m cos ω m t
2
dt
(4-2)
and the inertia force is
2
m s ⋅ v·· = – m s ⋅ v ⋅ ω m cos ω m t
Fig. 4.2—Lumped system for a foundation subjected to vertical, horizontal, and torsional excitation forces (Richart,
Hall, and Woods 1970).
(4-3)
where ms = effective mass of a spring.
In the absence of damping, the time-dependent relation of
the external harmonic force applied with amplitude F and
frequency ωo to the displacement v is
2
F ⋅ cos ω o t = k st ⋅ v ⋅ cos ω o t – m s ⋅ v ⋅ ω cos ω o t
(4-4)
and the amplitudes of force and displacement relate as
2
F = kst ⋅ v – ms ⋅ v ⋅ ω o
(4-5)
Fig. 4.3—Effect of mass on dynamic stiffness.
The dynamic stiffness, being the constant of proportionality
between the applied force and displacement, becomes
example vertical, is said to have only one degree of freedom
(Fig. 4.1(a)). The behavior of the mass depends on the nature
of both the spring and the dashpot.
The spring, presumed to be massless, represents the elasticity
of the system and is characterized by the stiffness constant k.
The stiffness constant is defined as the force that produces a
unit displacement of the mass. For a general displacement v
of the mass, the force in the spring (the restoring force) is kv.
In dynamics, the displacement varies with time t, thus, v =
v(t). Because the spring is massless, the dynamic stiffness
constant k is equal to the static constant kst, and k and the
restoring force kv are independent of the rate or frequency at
which the displacement varies.
The same concept of stiffness can be applied to a harmonically vibrating column that has its mass distributed along its
length (Fig. 4.3(a)). For an approximate analysis at low
frequency, the distributed column mass can be replaced by a
concentrated (lumped) mass ms, attached to the top of the
column, and the column itself can be considered massless
(Fig. 4.3(b)). Consequently, a static stiffness constant kst,
independent of frequency, can be used to describe the stiffness
of this massless column. The elastic force in the column just
2
k = kst – ms ω o
(4-6)
Thus, with vibration of an element having distributed mass,
the dynamic stiffness generally varies with frequency. At low
frequency, this variation is sometimes close to parabolic, as
shown on Fig. 4.3(c). The column used in this example may be
the column of soil and, thus, a soil deposit may feature stiffness
parameters that are frequency dependent. The magnitude and
character of this frequency effect depend on the size of the body,
vibration mode, soil layering, and other factors.
The dashpots of Fig. 4.2 are often represented as viscous
dampers producing forces proportional to the vibration
velocity v· of the mass. The magnitude of the damper force is
F D = cv· = c dv
-----dt
where
FD = damper force, lbf (N); and
c
= viscous damping constant, lbf-s/ft (N-s/m).
(4-7)
FOUNDATIONS FOR DYNAMIC EQUIPMENT
351.3R-31
The damping constant c is defined as the force associated
with a unit velocity. During harmonic vibrations, the timedependent viscous damping force is
cv· = – cvω m sin ω m t
(4-8)
and its peak value is cvωm. Equation (4-8) shows that for a
given constant c and displacement amplitude v, the amplitude of viscous damping force is proportional to frequency
(Fig. 3.8).
Several ways are used to account for variations of the
dynamic stiffness with frequency in the soil.
• Frequency-dependent equations—Veletsos and others
(Veletsos and Nair 1974; Veletsos and Verbic 1973; and
Veletsos and Wei 1971) have developed appropriate
equations that represent the impedance to motion offered
by uniform soil conditions. They developed these
formulations using an assumption of uniform elastic or
viscoelastic half-space and the related motion of a rigid
or flexible foundation on this half-space. Motions can be
translational or rotational. While the calculations can be
manually tedious, computer implementation provides
acceptable efficiency. Some of the simpler relevant
equations are presented in later sections of this report.
More extensive equations are presented in the references;
• Constant approximation—If the frequency range of
interest is not very wide, an often satisfactory technique
is to replace the variable dynamic stiffness by the constant
representative of the stiffness in the vicinity of the
dominant frequency (Fig. 4.3(c)). Another constant
approximation can be to add an effective or fictitious
in-phase mass of the supporting soil medium to the
vibrating lumped mass of the machine and foundation
and to consider stiffness to be constant and equal to the
static stiffness. Nevertheless, the variation of dynamic
stiffness with frequency can be represented by a
parabola only in some cases and the added mass is not
the same for all vibration modes; and
• Computer-based numerical analysis—For complex
geometry or variable soil conditions, the trend is to
obtain the stiffness and damping of foundations using
dynamic analysis of a three-dimensional or two-dimensional continuum representing the soil medium. The
continuum is modeled as an elastic or viscoelastic halfspace. The half-space can be homogeneous or nonhomogeneous (layered) and isotropic or anisotropic.
The governing equations are solved analytically or by
means of numerical methods such as the finite element
method. The major advantages of this elastic half-space
technique are that it accounts for energy dissipation
through elastic waves (geometric damping, also known
as radiation damping), provides for systematic analysis,
and uses soil properties defined by constants that can be
established by independent experiments. Computer
programs, such as DYNA, are very useful in this regard.
Commonly, the analysis of dynamically loaded foundations
is performed considering the rectangular foundation as if it
Fig. 4.4—Notation for calculation of equivalent radii for
rectangular bases.
were a circular foundation with equivalent properties (Fig. 4.4).
Some authors provide alternates to this approach (Bowles
1996). The equivalent circular foundation is not the same for
all directions of motion. For the three translational directions, the equivalent circular foundation is determined based
on an area equivalent to the rectangular foundation; thus,
these three share a common equivalent radius R. For the
three rotational directions, three unique radii are determined
that have moments of inertia equivalent to their rectangular
counterparts. These relationships are reflected in the
following equations
Translation: R =
ab
-----π
(4-9a)
3
Rocking: R ψ =
4
a b
-------- (about an axis parallel to b) (4-9b)
3π
2
Torsion: R η =
4
2
ab ( a + b )
---------------------------6π
(4-9c)
For rocking, different equivalent radii are determined for
each horizontal direction.
The equivalent radius works very well for square foundations
and rectangular foundations with aspect ratios a/b of up to 2.
With a/b ratios above 2, the accuracy decreases. For long
foundations, the assumption of an infinite strip foundation
may be more appropriate (Gazetas 1983).
These dynamics problems are addressed either with real
domain mathematics or with complex domain mathematics.
The real domain solution is more easily understood on a
subjective basis and is typified by the damped stiffness models
of Richart and Whitman (Richart and Whitman 1967; Richart,
Hall, and Woods 1970), in which the stiffness and damping
are represented as constants. The complex domain impedance
is easier to describe mathematically and is applied in the
impedance models of Veletsos and others (Veletsos and Nair
1974; Veletsos and Verbic 1973; Veletsos and Wei 1971).
Equation (4-10) characterizes the relationship between impedance models and damped stiffness models.
ki* = ki + iωmci
where
(4-10)
351.3R-32
ACI COMMITTEE REPORT
ki* = complex impedance for the i-th direction;
ki
= stiffness for the i-th direction;
–1 ;
i
=
ωm = circular frequency of motion; and
= damping constant for the i-th direction.
ci
Another common approach is to calculate impedance
parameters based on a dimensionless frequency ao computed as
ao = R ⋅ ωm ⁄ Vs = R ⋅ ωm ⋅ ρ ⁄ G
(1 – ν) m
0.425
Vertical: B v = ---------------- --------- and D v = ------------4 ρR 3
Bv
( 7 – 8ν ) -------m
0.288
Horizontal: B u = ---------------------- - and D u = ------------- (4-12b)
32 ( 1 – ν ) ρR 3
Bu
( 1 – ν ) I ψ and
Rocking: B ψ = 3------------------- ---------5
8
ρR ψ
(4-11)
where
R
=
equivalent foundation radius (Eq. (4.9a), (4.9b), or
(4.9c));
ωm = circular frequency of motion;
= shear wave velocity of the soil;
Vs
ρ
= soil mass density; and
G
= dynamic shear modulus of the soil.
In the most common circumstances, the soil and foundation
are moving at the same frequency the machine operates at.
Thus, ωm is commonly equal to ωo. In unusual cases such as
when motions result from a multiple harmonic excitation of
the equipment speed, from blade passage effects, from
multiple speed equipment combinations, and from some
poor bearing conditions, the motion may develop at a speed
different from the equipment speed. These situations are
more commonly found during in-place problem investigations rather than initial design applications. In those cases,
the correct frequency to consider is determined by in-place
field measurements.
4.2.1 Uniform soil conditions—In many instances, engineers
approximate or assume that the soil conditions are uniform
throughout the range of interest for the foundation being
designed. This assumption in some cases is reasonable. In
other cases, it is accepted for lack of any better model and
because the scope of the foundation design does not warrant
more sophisticated techniques. These are commonly referred
to as half-space models.
4.2.1.1 Richart-Whitman models—Engineers frequently
use a lumped parameter model (Richart and Whitman 1967),
which is considered suitable for uniform soil conditions.
This model represents the stiffness, damping, and mass for
each mode as singular, lumped parameters. In the norm,
these parameters are treated as frequency-independent. For
the vertical direction, the stiffness and damping equations
are expressly validated for the range of dimensionless
frequencies from 0 to 1.0 (0 < ao < 1.0) (Richart, Hall, and
Woods 1970). Implicitly, the other directions are similarly
limited, as the stiffness parameters are actually static stiffness values. Often, presentations of the data extend the range
to ao values as high as 1.5 to 2.0.
The mass of these models is determined solely as the translational mass and rotational mass moment of inertia for the
appropriate directions. No effective soil mass is included in
the representation (Richart, Hall, and Woods 1970).
The damping recommended by Richart (Richart, Hall, and
Woods 1970) associates with motion of rigid, circular foundations as given in the following equations
(4-12a)
(4-12c)
0.15
D ψ = -------------------------------( 1 + Bψ ) Bψ
Iη
0.50
Torsion: B η = --------- and D η = -----------------------5
(
+ 2B η )
1
ρR η
(4-12d)
where
Bi
=
=
Di
m
=
=
Ii
mass ratio for the i-th direction;
damping ratio for the i-th direction;
mass of the machine-foundation system;
mass moment of inertia of the machine-foundation
system for the i-th direction;
ρ
= soil mass density; and
R, Ri = effective radius of the foundation for the applicable
direction.
The lumped parameter models generally recognize two
alternate representations for the lumped stiffness terms.
These equations are based on the theory of elasticity for
elastic half-spaces with rigid foundations, except for horizontal motions, which make a slightly different rigidity
assumption. For a rigid circular foundation the applicable
formulas are given in Eq. (4-13a) to (4-13d). For a rectangular
foundation, the equivalent radii of Eq. (4-9) can be used in
the calculations. Alternatively, Eq. (4-13e) to (4-13g) can be
used directly for rectangular foundations of plan dimensions a
by b. In these equations, the βi values vary with the aspect
ratio of the rectangle. The βv value for the vertical direction
ranges from approximately 2.1 to 2.8, βu for the horizontal
ranges from approximately 0.95 to 1.2, and βψ for the
rocking varies from approximately 0.35 to 1.25. Specific
relationships can be found in a variety of sources (Richart,
Hall, and Wood 1970; Ayra, O’Neill, and Pincus 1979;
Richart and Whitman 1967), usually in a graphical form. The
original source materials contain the mathematical relationships. There is little difference between using either the circular
or the rectangular formulations for these stiffness terms.
4
Vertical: k v = ----------------GR
(1 – ν)
(4-13a)
32 ( 1 – ν )
Horizontal: k u = ----------------------- GR
( 7 – 8ν )
(4-13b)
FOUNDATIONS FOR DYNAMIC EQUIPMENT
3
8
Rocking: k ψ = -------------------- GR ψ
3(1 – ν)
(4-13c)
3
16
Torsion: k η = ------ GR η
3
(4-13d)
G
Vertical: k v = ----------------β v ab
(1 – ν)
(4-13e)
Horizontal: k u = 2 ( 1 + ν )Gβ u ab
(4-13f)
2
G
Rocking: k ψ = ----------------β ψ ba
(1 – ν)
(4-13g)
2
β1 ( β2 ao )
χ ψ = --------------------------2
1 + ( β2 ao )
2
ψψ
2
4GR
k v∗ = ---------------- [ ( 1 – χ v – γ 3 a o ) + ia o ( γ 4 + ψ v ) ]
(1 – ν)
where
2
stiffness for the i-th direction;
shear modulus of the soil;
equivalent foundation radius;
Poisson’s ratio of the soil;
plan dimensions of a rectangular foundation; and
rectangular footing coefficients (Richart, Hall,
and Woods 1970).
The damping constants can be determined from the
computed damping ratio, system mass, and stiffness as
damping constant for the i-th direction;
damping ratio for the i-th direction;
stiffness for the i-th direction;
mass of the machine-foundation system; and
mass moment of inertia of the machine-foundation system for the i-th direction.
4.2.1.2 Veletsos models—For the impedance functions
of foundations resting on the surface of a viscoelastic halfspace, Veletsos and Verbic (1973) determined the analytical
expressions for impedance dependent on frequency, Poisson’s
ratio, and internal material damping. Neglecting the internal
material damping, the relationships are
Horizontal impedance:
(4-15a)
Rocking impedance
3
where
2
γ1 γ2 ⋅ ( γ2 ao )
ψ v = ------------------------------2
1 + ( γ2 ao )
3
16GR
Torsional impedance: k η∗ = ----------------η- [ A + ia o B ] (4-15d)
3
where
2
b1 ⋅ ( b2 ao )
A = 1 – --------------------------2
1 + ( b2 ao )
(4-14)
where
=
ci
=
Di
ki
=
m
=
=
Ii
8GR ψ
2
[ ( 1 – χ ψ – β 3 a o ) + ia o ψ ψ ]
k ψ∗ = ------------------3(1 – v)
(4-15c)
γ1 ⋅ ( γ2 ao )
χ v = -------------------------2
1 + ( γ2 ao )
where
=
ki
G
=
R, Ri =
ν
=
a, b =
=
βi
8GR
k u∗ = ----------- [ 1 + ia o α 1 ]
2–v
β1 β2 ⋅ ( β2 ao )
= --------------------------------2
1 + ( β2 ao )
Vertical impedance
(about an axis parallel to b)
c i = 2D i k i m or c i = 2D i k i I i
351.3R-33
(4-15b)
2
b1 b2 ⋅ ( b2 ao )
B = -------------------------------2
1 + ( b2 ao )
where
ki*
R, Ri
G
ao
ν
α1, βj , and γj
=
=
=
=
=
=
impedance in the i-th direction;
equivalent foundation radius;
dynamic shear modulus of the soil;
dimensionless frequency;
Poisson’s ratio of the soil;
coefficients dependent on Poisson’s ratio
as given in Table 4.1;
j
= 1 to 4 as appropriate; and
= 0.425 and 0.687, respectively (Veletsos
b1, b2
and Nair 1974).
The leading terms of each of these impedance equations are
the static stiffnesses of the foundation for motion in that
direction. For three cases (vertical, rocking, and torsional
motion), these terms are the same as presented in the RichartWhitman lumped parameter model. For the horizontal motion,
there is a slight difference due to assumptions about the
foundation rigidity. The practicality of the difference is small,
especially toward the higher range of Poisson’s ratio values.
These equations may also be expressed as
351.3R-34
ACI COMMITTEE REPORT
Table 4.1—Values of α1, βj, and γj
ν=0
ν = 0.33
ν = 0.45
ν = 0.50
α1
0.775
0.650
0.600
0.600
β1
0.525
0.500
0.450
0.400
β2
0.800
0.800
0.800
0.800
β3
0.000
0.000
0.023
0.027
γ1
0.250
0.350
—
0.000
γ2
1.000
0.800
—
0.000
γ3
0.000
0.000
—
0.170
γ4
0.850
0.750
—
0.850
Table 4.2—Stiffness and damping parameters
(D = 0)
Motion
Horizontal
Rocking
Torsion
Vertical
Soil
Side layer
Half space
Cohesive
Su1 = 4.1
Su2 = 10.6
Cu1 = 5.1
Granular
Su1 = 4.0
Su2 = 9.1
Cu1 = 4.7
Cu2 = 3.2
Cu2 = 2.8
Cohesive
Sψ1 = 2.5
Sψ2 = 1.8
Cψ1 = 4.3
Cψ2 = 0.7
Granular
Sψ1 = 2.5
Sψ2 = 1.8
Cψ1 = 3.3
Cψ2 = 0.5
Cohesive
Sη1 = 10.2
Sη2 = 5.4
Cη1 = 4.3
Cη2 = 0.7
Granular
Sη1 = 10.2
Sη2 = 5.4
Cη1 = 4.3
Cη2 = 0.7
Cohesive
Sv1 = 2.7
Sv2 = 6.7
Cv1 = 7.5
Cv2 = 6.8
Granular
Sv1 = 2.7
Sv2 = 6.7
Cv1 = 5.2
Cv2 = 5.0
Note: S values are valid for 0.5 < ao < 1.5; C values for valid for ao < 2.0.
Horizontal impedance: ku* = GR(Cu1 + iaoCu2)
(4-16a)
Rocking impedance: kψ* = GRψ3 (Cψ1 + iaoCψ2) (4-16b)
Vertical impedance: kv* = GR(Cv1 + iaoCv2)
(4-16c)
Rocking impedance: kη* = GRη3 (Cη1 + iaoCη2) (4-16d)
where
ki*
= impedance in the i-th direction;
R, Ri
= equivalent foundation radius;
G
= dynamic shear modulus of the soil; and
= dimensionless parameters.
Ci1, Ci2
Constant approximations of Ci1 and Ci2 are given in
Table 4.2 for two broad classes of soils (cohesive and granular)
and for dimensionless frequency values ao less than 2.0. Polynomial expansions of these approximations are also available to
cover a wider range of dimensionless frequencies.
4.2.1.3 Other models—Engineers have used finite
element models as another approach to obtain stiffness and
damping relationships. Rather than theoretically assessing
the behavior of an elastic continuum, parametric studies with
a finite element model determine the impedance relationships. Given all the approximations involved, the agreement
between the solutions—the half-space theory and the finite
element modeling—is very good. The exception is the
rocking stiffness kψ for which researchers have computed
substantially larger values by finite element solutions
(Karabalis and Beskos 1985; Kobayashi and Nishimura
1983; Wolf and Darbre 1984). The empirical expressions for
the static stiffness of circular foundations embedded into a
homogeneous soil layer can be found in Elsabee and Morray
(1977) and Kausel and Ushijima (1979).
4.2.2 Adjustments to theoretical values—Damping in a
soil-foundation system consists of two components:
geometric damping and material damping. The geometric
damping, also called radiation damping, represents a
measure of the energy radiated away from the immediate
region of the foundation by the soil. Material damping
measures the energy loss as a result of soil hysteresis effects.
Sections 4.2.1.1 and 4.2.1.2 present equations to evaluate
geometric damping that can be evaluated for various shape
foundations and different types of soils using elastic halfspace theories. Both experience and experimental results
show that damping values for large foundations undergoing
small vibration amplitudes are typically less than those
analytically predicted values (EPRI 1980; Novak 1970). This
discrepancy is attributed to the presence of soil layers that
reflect waves back to the vibrating foundation. Various sources
recommend reduced soil values (EPRI 1980; Novak 1970; DIN
4024; and Gazetas 1982). Specific recommendations vary with
the type of application. The foundation engineer needs to select
proper soil damping values and limits based on the specific
application. For example, EPRI 1980 recommends the soil
damping ratio for use in the design of power plant fan
foundations should not exceed 20% for horizontal motion,
50% for vertical motion, 10% for transverse rocking motion,
and 15% for axial and torsional motions. German DIN 4024
recommends that the soil damping ratios used in vibration
analysis of rigid block foundations should not exceed 25%.
Novak (1970) recommends reducing the analytically determined geometric damping ratios (from elastic half-space
models) by 50% for a dynamic analysis of the foundation.
Although material damping is often neglected, as
presented in Section 4.2.1, that assumption often leads to
overestimating the first resonant amplitude of the coupled
translation/rocking response of surface footings by several
hundred percent because very limited geometric damping
develops during rocking. This overestimation can be reduced
by the inclusion of material damping (Section 4.2.4).
Torsional response is difficult to predict because of slippage.
For surface foundations, slippage reduces stiffness and
increases damping; for embedded foundations, slippage
reduces damping. The inclusion of the weakened zone
around the footing may improve the agreement between the
theory and the experiments. This typically requires more
complete computer-based numerical analysis.
Another correction of the half-space theory may be
required if the soil deposit is a shallow layer. In such cases,
the stiffness increases and geometric damping decreases and
can even vanish if the frequency of interest (for example, the
excitation frequency) is lower than the first natural frequency
of the soil layer (Kausel and Ushijima 1979). For a homogeneous layer of thickness H with soil shear wave velocity Vs,
the first natural frequencies of the soil deposit are
FOUNDATIONS FOR DYNAMIC EQUIPMENT
πV 2 ( 1 – ν )
Horizontal direction: ω su = --------S- -------------------2H 1 – 2ν
(4-17)
πV
Vertical direction: ω sv = --------S2H
(4-18)
where
ωsu, ωsv = circular natural frequencies of a soil layer in u
and v directions;
Vs
= soil shear wave velocity;
H
= depth of soil layer; and
ν
= Poisson’s ratio of the soil.
At excitation frequencies ωo lower than ωsu and ωsv, only
material damping remains because no progressive wave
occurs to generate geometric damping in the absence of
material damping, and only a very weak progressive wave
occurs in the presence of material damping. The damping
parameters generated by the material alone to be used for
excitation frequencies ωo lower than ωsu and ωsv are
cu = 2βm ⋅ ku /ωo
(4-19)
cv = 2βm ⋅ kv /ωo
(4-20)
where βm = material damping ratio, and other terms are as
previously defined. In the impedance models, the imaginary
terms of the horizontal (ku*) and vertical (kv*) impedance
become +2βmi within the brackets of Eq. (4-15a) and (4-15c).
This correction is most important for the vertical and horizontal directions, in which the geometric damping of the
half-space is high but is minimal for the shallow layer. This
correction is not an absolute breakpoint based on the
computed layer frequency and the excitation frequency.
Short of using computer-based numerical solution techniques that reasonably represent the loss of geometric
damping, the foundation engineer should apply judgement to
decrease the geometric damping for shallow layer sites.
4.2.3 Embedment effects—Most footings do not rest on the
surface of the soil but are partly embedded. Studies have
shown that embedment increases both stiffness and
damping, but the increase in damping is more significant.
Overall, embedment effects are often overestimated
because soil stiffness (shear modulus) diminishes toward the
soil surface due to diminishing confining pressure. This is
particularly so for backfill lacking a stiff surface crust and
whose effects are always much less pronounced than those
of undisturbed soil. The lack of confining pressure at the
surface often leads to separation of the soil from the foundation
and to the creation of a gap as indicated on Fig. 4.5, which
significantly reduces the effectiveness of embedment. To
find an approximate correction for this effect, the engineer
should consider an effective embedment depth less than the
true embedment.
In determining the stiffness of embedded foundations, an
approximate but versatile approach was formulated using the
assumption that the soil reactions acting on the base can be
351.3R-35
Fig. 4.5—Schematic of embedded foundation.
taken as equal to those of a surface foundation (half-space)
and the reactions acting on the footing sides as equal to those
of an independent layer overlying the half-space (Fig. 4.5).
The evaluation of the reactions of a layer is simplified and
calculated using the assumption of plane strain. This means
that these reactions are taken as equal to those of a rigid,
infinitely long, massless cylinder undergoing a uniform
motion in an infinite homogeneous medium.
The plane strain approach to the side reactions has many
advantages: it accounts for energy radiation through wave
propagation, leads to closed-form solutions, and allows for
the variation of the soil properties with depth. It can also
allow for the slippage zone around the footing. Also, the
approach is simple and makes it possible to use the solutions
of surface footings because the effect of the independent side
layer actually represents an approximate correction of the
half-space solutions for the embedment effect. This
approach works quite well, and its accuracy increases with
increasing frequency.
Equation (4-21a) to (4-21d) describe the side resistance of
the embedded cylinder, analogous to the surface disc, using
complex, frequency-dependent impedance
Horizontal impedance: k eu∗ = G s l [ S u1 + i ⋅ a o ⋅ S u2 ] (4-21a)
Vertical impedance: k ev∗ = G s l [ S v1 + i ⋅ a o ⋅ S v2 ] (4-21b)
Rocking impedance:
(4-21c)
k e ψ∗ = G s R ψ l [ S ψ1 + i ⋅ a o ⋅ S ψ2 ]
2
Torsional impedance:
(4-21d)
k e η∗ = G s R η l [ S η1 + i ⋅ a o ⋅ S η2 ]
2
where
kei*
Gs
=
=
impedance in the i-th direction due to embedment;
dynamic shear modulus of the embedment
(side) material;
351.3R-36
l
Ri
=
=
ACI COMMITTEE REPORT
depth of embedment (effective);
equivalent foundation radius (Eq. (4.9b) or
(4.9c)); and
Si1, Si2 = dimensionless parameters (Table 4.2).
In these expressions, the shear modulus Gs is that of the
side layer that may represent the backfill. The dimensionless
parameters Si1 and Si2 relate to the real stiffness and the
damping (out-of-phase component of the impedance),
respectively. These parameters depend on the dimensionless
frequency ao (Eq. (4-11)) applicable for the layer of embedment
material. Poisson’s ratio affects only the horizontal impedance
generated by the footing embedment, not the impedance in
other directions. In complete form, these Si parameters also
depend on material damping of the side layer soils.
The mathematical expressions for the parameters Si1 and
Si2 can be found in Novak, Nogami, and Aboul-Ella (1978)
and Novak and Sheta (1980). These parameters are frequency
dependent; nevertheless, given all the approximations involved
in the modeling dynamic soil behavior, it is often sufficient
to select suitable constant values to represent the parameters,
at least over limited frequency range of interest. Table 4.2
shows constant values for cohesive soils and granular soils
with Poisson’s ratio of 0.4 and 0.25, respectively. The values
correspond to dimensionless frequencies between 0.5 and
1.5, which are typical of many machine foundations. If a
large frequency range is important, parameter S should be
considered as frequency dependent and calculated from the
expression of impedance functions given in Novak, Nagami,
and Aboul-Ella (1978) and Novak and Sheta (1980). Material
damping is not included in Table 4.2 but can be accounted
for by using Eq. (4-23).
The engineer can approximate the complex stiffness of
embedded foundations by adding the stiffness generated by
the footing sides to that generated in the base. In some cases,
consideration of the difference in location of the embedment
impedance and the basic soil impedance may be included in
the analysis. For vertical translation and torsion, the total
stiffness and damping results in simple addition of the two
values. For horizontal translation and rocking, coupling
between the two motions should be considered.
4.2.4 Material damping—Material damping can be incorporated into the stiffness and damping of the footing in
several ways. The most direct way is to introduce the
complex shear modulus into the governing equations of the
soil medium at the beginning of the analysis and to carry out
the whole solution with material damping included.
Another way is to carry out the purely elastic solution and
then introduce material damping into the results by applying
the correspondence principle of viscoelasticity. With steadystate oscillations considered in the derivation of footing
stiffness, the application of the correspondence principle
consists of replacing the real shear modulus G by the
complex shear modulus G*.
This replacement should be done consistently wherever
G occurs in the elastic solution. This includes the shear
wave velocity (Eq. (3-32)) and the dimensionless
frequency (Eq. (4-11)), which consequently become
complex. Therefore, all functions that depend on ao are
complex as well. The substitution of G* can be done if
analytical expressions for the impedance ki* or constants ki
and ci are available from the elastic solution. With the material
damping included, the parameters have the same meaning as
before but also depend on the material damping.
The aforementioned procedures for the inclusion of material
damping into an elastic solution are accurate but not always
convenient. When the elastic solution is obtained using a
numerical method, the impedance functions are obtained in
a digital or graphical form, and analytical expressions are not
available. In such cases, an approximate approach often used
to account for material damping multiplies the complex
impedance, evaluated without regard to material damping,
by the complex factor (1+ i2βm) to determine an adjusted
complex impedance
ki*(adj) = (ki + iωm ⋅ ci ) ⋅ (1 + 2 ⋅ βm ⋅ i)
(4-22)
= (ki – 2 ⋅ βm ⋅ ωm ⋅ ci) + iωm ⋅ (ci + 2 ⋅ βm ⋅ ki /ωm)
where βm = material damping ratio of the soil, and other
terms are as previously defined.
Recognizing the stiffness as the real part of the impedance
and the damping as the imaginary term of the impedance, the
adjusted stiffness and damping terms considering material
damping become
ki (adj) = ki – 2 ⋅ βm ⋅ ci ⋅ ωm and
(4-23)
ci(adj) = ci + 2 ⋅ βm ⋅ ki /ωm
where ki and ci are calculated assuming perfect elasticity,
and ci includes only geometric damping. Studies indicate
that this approximate approach gives sufficient accuracy at
low dimensionless frequencies, but the accuracy deteriorates
with increasing frequency. Equation (4-22) and (4-23) show
that material damping reduces stiffness in addition to
increasing damping.
As another approach, variations of Eq. (4-15) are available,
which include material damping directly as a distinct parameter
(Veletsos and Verbic 1973; Veletsos and Nair 1974).
Finally, some engineers simply add the material damping
to the geometric damping otherwise determined by the
preceding equations. This approach is more commonly used
with the Richart-Whitman formulations and does not alter
the stiffness. In such circumstances, broad judgements are
often applied at the same time so that if the geometric
damping is large, the material damping may be neglected.
Similarly, material damping may be included only in those
cases where excessive resonance amplification is expected.
This simple additive approach is generally recognized as the
least accurate of the possible methods.
4.2.5 Pile foundations—Stiffness and damping of piles are
affected by interaction of the piles with the surrounding soil.
In the past, consideration of this interaction was limited to
the determination of the length of the so-called equivalent
FOUNDATIONS FOR DYNAMIC EQUIPMENT
351.3R-37
Fig. 4.6—Mathematical models used for dynamic analysis
of piles.
cantilever, a freestanding bare pile shorter than the embedded
pile. Pile damping was estimated.
The soil-pile interaction under dynamic loading modifies
the pile stiffness, making it frequency-dependent. As with
shallow foundations, this interaction also generates
geometric damping. In groups of closely spaced piles, the
character of dynamic stiffness and damping is further
complicated by interaction between individual piles known
as pile-soil-pile interaction or group effect.
Therefore, recent approaches for determining stiffness and
damping of piles consider soil-pile interaction in terms of
continuum mechanics and account for propagation of elastic
waves. The problem solutions are based on a few
approaches, such as the continuum approach, the lumped
mass model, the finite element model, and the boundary
integral method (Fig. 4.6).
In most cases, the impedance provided to a pile foundation
is determined considering only the effects of the piles.
Because piles are typically used due to poor surface layer
soils, the effects of the soils directly under the cap are often
neglected; a settlement gap is assumed to develop. Similarly,
the effects of embedment are often neglected. If circumstances
indicate that the embedment effect may be significant, the
procedures outlined in Section 4.2.3 are often applied.
The basic approach toward pile analysis is to first evaluate
the characteristics of a single pile. Once these parameters
(stiffness and damping) are established for the single pile,
the group effects are determined. Other approaches, such as
finite element analysis, may model and consider both effects
simultaneously.
4.2.5.1 Single piles—Dynamic behavior of embedded
piles depends on frequency and the properties of both the pile
and soil. The pile is described by its length, bending and axial
stiffnesses, tip/end conditions, mass, and batter (inclination
from the vertical). Soil behavior depends on soil properties
and the soil’s variation with depth (layering).
Dynamic response of a pile-supported foundation depends
on the dynamic stiffness and damping of the piles. These
properties for a single pile can be described in terms of either
impedance or stiffness and equivalent viscous damping. As
previously established, these are related as
ki*
= ki + iωmci
where
ki* = complex impedance for the i-th direction;
(4-24)
Fig. 4.7—Generation of pile stiffness in individual directions.
ki = stiffness for the i-th direction;
i
= –1 ;
ωm = circular frequency of motion; and
ci = damping constant for the i-th direction.
The engineer can determine the constants experimentally
or theoretically. The theoretical approach is commonly used
because experiments, though very useful, are difficult to
generalize. In the theoretical approach, dynamic stiffness is
generated by calculating the forces needed to produce a unit
amplitude vibration of the pile head in the prescribed direction
(Fig. 4.7).
For a single pile, the impedance at the pile head can be
determined from the following
Ep Ap
Vertical translation: k vj = -----------f
r o v1
(4-25a)
Ep Ap
and c vj = -----------f
V S v2
Ep Ip
-f
Horizontal translation: k uj = --------3 u1
ro
(4-25b)
Ep Ip
- f u2
and c uj = ---------2
ro VS
Rotation of the pile head in the vertical plane:
Ep Ip
Ep Ip
-f ψ1 and c ψj = ---------f
k ψj = --------ro
V S ψ2
(4-25c)
Coupling between horizontal translation and rotation:
Ep Ip
Ep Ip
-f
-f
k uψj = --------and c uψj = --------2 uψ1
r o V s uψ2
R
(4-25d)
Gp J
Gp J
-f η1 and c nj = --------f
Torsion: k nj = -------ro
V s η2
(4-25e)
where
= stiffness of pile j in the i-th direction;
kij
351.3R-38
cij
ACI COMMITTEE REPORT
= equivalent viscous damping of pile j in the i-th
direction;
= Young’s modulus of the pile;
Ep
= cross-sectional area of the pile;
Ap
= moment of inertia of the pile cross section;
Ip
= pile radius or equivalent radius;
ro
Gp J = torsional stiffness of the pile;
= shear wave velocity of the soil; and
Vs
fi1 , fi2 = dimensionless stiffness and damping functions
for the i-th direction.
Graphical or tabular presentations of the fi1 and fi2 functions
are presented in a variety of sources (Novak 1974, 1977;
Kuhlmeyer 1979a,b) and are included in some software
packages. If the pile heads are pinned into the foundation
block, then kψ = kuψ = kη = 0 and cψ = cuψ = cη = 0 in the
previous formulas, and ku should be evaluated for pinned
head piles. The vertical constants labeled v are the same for
the fixed and pinned heads. The rotational parameters (fψ1,
fψ2, fuψ1, fuψ2, fη1, and fη2) are applicable only if the pile is
assumed or designed to be rotationally fixed to the pile cap.
In general, the fi1 and fi2 functions depend on the following
dimensionless parameters:
• Dimensionless frequency ao = ωmro /Vs (note that this
value is calculated using the pile radius and is a much
smaller value than the ao calculated for a complete soil
supported foundation);
• Relative stiffness of the soil to the pile, which can be
described either by the modulus ratio G/Ep or the
velocity ratio Vs /Vc in which Vc is the compression
wave velocity of the pile equal to E p ⁄ ρ p with ρp equal
to the pile mass density;
• The mass ratio ρ/ρp of the soil and the pile;
• The slenderness ratio lp/ro in which lp is pile length;
• Material damping of both soil and pile;
• The pile’s tip restraint condition and rotational fixity of
the head; and
• Variation of soil and pile properties with depth.
These factors affecting the functions f are not of equal
importance in all situations. Often, some of them can be
neglected, making it possible to present numerical values of
function f in the form of tables and charts for some basic cases.
The pile stiffness diminishes with frequency quickly if the
soil is very weak relative to the pile. This happens when the
soil shear modulus is very low or when the pile is very stiff.
In addition, dynamic stiffness can be considered practically
independent of frequency for slender piles in average soil.
The imaginary part of the impedance (pile damping)
grows almost linearly with frequency and, therefore, can be
represented by constants of equivalent viscous damping ci ,
which are also almost frequency independent. Only below
the fundamental natural frequencies of the soil layer Eq. (4-17)
and (4-18) does geometric damping vanish and material
damping remain as the principal source of energy dissipation.
Then soil damping can be evaluated using Eq. (4-19) and
(4-20). The disappearance of geometric damping may be
expected with low frequencies and shallow layers, stiff soils,
or both. Apart from these situations, frequency independent
viscous damping constants, and functions fi2, which define
them, are sufficient for practical applications.
The mass ratio ρ/ρp is another factor whose effect is
limited to extreme cases. Only for very heavy piles do the
pile stiffness and damping change significantly with the
mass ratio.
The Poisson’s ratio effect is very weak for vertical vibration,
absent for torsion, and not very strong for the other mode of
vibration, unless the Poisson’s ratio approaches 0.5 and
frequencies are high. The effect of Poisson’s ratio on parameters fi1 and fi2 can be further reduced if the ratio E/Ep rather
than G/Ep is used to define the stiffness ratio.
The slenderness ratio, lp/ro, and the tip conditions are very
important for short piles, particularly for vertical motion
because the piles are stiff in that direction. Floating piles
(also called friction piles) have lower stiffness but higher
damping than end bearing piles. In the horizontal direction,
piles tend to be very flexible. Consequently, parameters fi1
and fi2 become practically independent of pile length and the
tip condition for ratios lp /ro greater than 25 if the soil
medium is homogeneous.
Observations suggest that the most important factors
controlling the stiffness and damping functions fi1 and fi2 are
the stiffness ratio relating soil stiffness to pile stiffness, the soil
profile, and, for the vertical direction, the tip restraint condition.
4.2.5.2 Pile groups—Piles are usually used in a group.
The behavior of a pile group depends on the distance between
individual piles. When the distance between individual piles is
large—20 diameters or more—the piles do not affect each
other, and the group stiffness and damping are the sums of
the contributions from the individual piles. If, however, the
piles are closely spaced, they interact with each other. This
pile-soil-pile interaction or group effect exerts a considerable influence on the stiffness and damping of the group.
These two basic situations may be treated separately.
Pile interaction neglected—When spacing between piles
reaches 20 diameters or more, the interaction between piles
can be neglected. Then stiffness and damping of the pile
group can be determined by the summation of stiffness and
damping constants of the individual piles. In many cases,
initial calculations are performed neglecting the interaction.
An overall group efficiency factor is then determined and
applied to the summations.
In the vertical and horizontal directions, the summation is
straightforward. For torsion and sliding coupled with
rocking, the position of the center of gravity CG and the
arrangement of the piles in plan are important. Thus, the
group stiffness and damping with respect to rotation derive
from the horizontal, vertical, and moment resistance of individual piles and the pile layout.
As the rigid body is rotated, an amount ψ at the CG (Fig. 4.8),
the head of pile j undergoes horizontal translation uj =ψyj,
vertical translation vj =ψxj, and rotation ψj = ψ. For the
torsional stiffness and damping of the group, the twist η
applied at the CG twists the pile by the same angle and
translates its head horizontally by the distance equals to
2
2
η x j + z j (Fig. 4.9). With these considerations and notations
shown on Fig. 4.8 and 4.9, the stiffness and damping constants
FOUNDATIONS FOR DYNAMIC EQUIPMENT
351.3R-39
of the pile group for individual motions as referenced to the
centroid of the pile group are as follows
N
N
∑
Vertical translation: k gv =
k vj and c gv =
j=1
∑ cvj
(4-26a)
j=1
Horiztonal translation:
N
k gu =
∑
N
k uj and c gu =
j=1
∑ cuj
(4-26b)
j=1
Rotation of the cap in the vertical plane:
N
∑ ( kψj + kvj xj ) and
2
kg ψ =
(4-26c)
j=1
Fig. 4.8—Pile displacements for determination of group
stiffness and damping related to unit rotation ψ.
N
∑ ( cψj + cvj xj )
2
cg ψ =
j=1
Coupling between horizontal translation and rotation:
N
k gu ψ = k g ψu =
∑ ku ψj and
(4-26d)
j=1
N
c gu ψ = c g ψu =
∑ cu ψj
j=1
Torsion about vertical axis:
(4-26e)
Fig. 4.9—Pile displacements for determination of group
stiffness and damping related to unit torsion η.
where
kgi = pile group stiffness in the i-th direction; and
cgi = pile group damping in the i-th direction.
The summations extend over all the piles. The distances xj
and zj refer to the distances from the centroid of the pile
group to the individual pile. If the CG is located directly
above the pile group centroid, these distances are as indicated in Fig. 4.8 and 4.9. The vertical eccentricity yc must be
addressed as presented in Section 4.2.6. These stiffness and
damping terms, or their impedance equivalents, represent
values comparable to the terms developed for a soilsupported foundation in Section 4.2.1.
Pile interaction considered—When piles are closely
spaced, they interact with each other because the displacement
of one pile contributes to the displacements of others.
Studies of these effects call for the consideration of the soil
as a continuum.
For static loads, these studies indicate that the main results
of the static pile interaction are an increase in settlement of
the group, a redistribution of the pile stresses, and, with rigid
caps, a redistribution of pile reactions.
The studies of the dynamic pile-soil-pile interaction
suggest a number of different observations:
• Dynamic group effects are profound and differ considerably from static group effects;
• Dynamic stiffness and damping of piles groups vary
with frequency, and these variations are more dramatic
than with single piles; and
• Group stiffness and damping can be either reduced or
increased by pile-soil-pile interaction.
These effects can be demonstrated if the group stiffness
and damping are described in terms of the group efficiency
ratio GE defined as
N
kg η =
∑ [ knj +
2
k uj ( x j
+
2
zj ) ]
and
j=1
N
cg η =
∑ [ cnj + cuj ( xj + zj ) ]
2
2
j=1
351.3R-40
ACI COMMITTEE REPORT
Group efficiency =
(4-27)
k group
Group stiffness
------------------------------------------------------------------------------------ = GE = ------------Sum of stiffness of individual piles
kr
∑r
where kr is the stiffness of individual pile considered in
isolation. When the pile-soil-pile interaction effects are
absent, GE = 1. The group efficiency of damping can be
defined in the same way.
Dynamic group effects are complex, and there is no simple
way of alleviating these complexities. The use of suitable
computer programs appears necessary to describe the
dynamic group stiffness and damping over a broad
frequency range (Novak and Sheta 1982).
The principle alternatives available are replacing the pile
group by an equivalent pier, equating dynamic interaction to
the static interaction factors, or using the dynamic interaction
factors. The equivalent pier may only be applicable for very
closely spaced piles and may overestimate damping. The
static interaction may be sufficiently accurate for dynamic
analysis if the frequencies of interest are low, especially if
these frequencies are lower than the natural frequencies of the
soil deposit as determined by Eq. (4-17) and (4-18).
Pile group stiffness using static interaction coefficients—
An accurate analysis of static behavior of pile groups should
be performed using a suitable computer program (Sharnouby
and Novak 1985; Poulos and Randolph 1983). Nevertheless,
a simplified approximate analysis suitable for hand calculations
can be formulated based on interaction factors α. The interaction factors derive from the deformations of two equally
loaded piles and give the fractional increase in deformation
of one pile due to the deformation of an equally loaded adjacent
pile. The flexibility and stiffness are then established by
superposition of the interaction between individual pairs of
piles in the group. The approximation comes from neglecting
the stiffening effects of the other piles when evaluating the
factor α. The accuracy of the approach appears adequate, at
least for small to moderately large groups. Poulos and Davis
(1980) contains charts for the interaction factors for both
axial and lateral loading.
The ratio ρa = Gave /Gl accounts for the variation of soil
stiffness with depth, where Gave is the average shear
modulus along the length of the pile, and Gl is the shear
modulus at the pile tip. The stiffness of the pile, relative to
the soil, is defined by the stiffness ratio λ = Ep/Gl. An
approximately linear relationship exists between interaction
factors in the vertical direction αv and ln(s/d) (Poulos 1979).
For λ ≅ 500, typical of offshore structures, αv can be estimated
from the following formula (Randolph and Poulos 1982)
ln ( l p ⁄ s )
for s ≤ lp
α v = 0.5 ----------------------ln ( l p dρ a )
where
αv =
lp
=
(4-28)
vertical interaction coefficient between two piles;
pile length;
s
= distance between piles;
d
= pile diameter; and
ρa = Gave /Gl.
For lateral loading, the pile behavior depends on the length
of that part of the pile that deforms appreciably under lateral
loading. This critical length may be estimated from the
formula (Randolph 1981)
lc = 2ro(Ep/Gc)2/7
(4-29)
where
lc = critical length of a pile;
ro = pile radius;
Ep = Young’s modulus of the pile; and
Gc = the average value of dynamic shear modulus of the
soil over the critical length.
A few iterations may be needed to find corresponding
values of Gc and lc.
The interaction factors for horizontal translation u and
rotation ψ may be estimated as (Randolph and Poulos 1982)
αuf = 0.6ρc(Ep/Gc)1/7(ro /s)(1 + cos2β)
(4-30a)
αuH = 0.4ρc(Ep/Gc)1/7(ro /s)(1 + cos2β)
(4-30b)
2
3
αψH = α uH and αψM = α uH
(4-30c)
where
ρc =
Gz =
αuf =
Gz /Gc
the shear modulus at depth z = lc /4;
the horizontal interaction factor for fixed-headed
piles (no head rotation);
αuH = the horizontal interaction factor due to horizontal
force (rotation allowed);
αψH = the rotation due to horizontal force;
αψM = the rotation due to moment; and
βp = the angle between the direction of the loading and
the line connecting the pile centers (Fig. 4.10).
When the calculated interaction factor α exceeds 1/3, its
value should be replaced by
2
α′ = 1 – -------------27α
(4-31)
This correction is made to avoid α approaching infinity as
s approaches 0.
For the vertical stiffness of a symmetrical pile group,
assuming that all piles carry the same load, the group stiffness may be estimated as
∑j kvj
k gv = ------------αv
∑j
(4-32)
FOUNDATIONS FOR DYNAMIC EQUIPMENT
in which kvj is the vertical stiffness of a single pile, and αv is
the interaction factor between a reference pile i and any other
pile in the group. The summation in the denominator then
represents a summation of factors between the reference pile
and all other piles in the group. The reference pile should not
be in the center or at the periphery and has an interaction
factor with itself of αv = 1.
If a rigid cap is assumed, which implies the same displacements for all piles, the piles have different individual stiffness
values and the following formula is applied (Novak 1979)
k gv = k vj
∑i ∑r εir
(4-33)
where
εir =
[αir] =
the elements of the inverted matrix [αir]–1; and
matrix of factors between any two piles with diagonal terms αii = 1.
The difference between the two previous formulas for kgv
is usually not great. For horizontal stiffness, the approximate
correction may be done in a similar fashion using factors αuf
or αuH.
For rotation of a thin rigid cap, the rocking stiffness comes
primarily from the vertical stiffness of the piles. This part of
the group stiffness becomes
kg ψ = kv
∑i ∑r εir xi xr
(4-34)
in which x is the horizontal distance of the pile from the axis
of rotation. For thick caps, these corrections can be introduced into the equations of the rotation of the cap in the
vertical plane—the case where pile interaction is neglected.
For torsion of the cap, ignoring the contribution from
individual pile twisting, the group stiffness can be written
analogously as
k ηη = k u 

∑i ∑r εir ( x) zi zr + ∑i ∑r εir ( z ) xi xr
(4-35)
in which x and z are the pile coordinates indicated in Fig. 4.9.
If αir(x) and αir(z) are horizontal interaction factors between
piles i and r in direction X and Z, respectively, εir are the
elements in [εir] = [αir] –1.
The static procedure does not offer any guidance as to the
effect of interaction on group damping. Group interaction
usually increases the damping ratio, not necessarily the damping
constant c. To account for this approximately, the group
damping constants may be taken as cg ≅
Fig. 4.10—Definitions of s and βp.
Dynamic displacement of Pile 2
----------------------------------------------------------------------------Static displacement of Pile 1
in which the displacement of Pile 2 is caused by a unit
harmonic load of Pile 1, and the static displacement of Pile 1 is
established for an isolated pile. The displacement means either
a translation or rotation. These dynamic interaction factors are
used in association with stiffness and damping of single piles
in the same way that static interaction factors are used.
The use of dynamic interaction factors yields complex
group stiffness kg* = kg + ik2 whose imaginary part defines
the group damping constant cg = k2 /ωm. An increase in
damping and strong variation with frequency is often
obtained. Mitwally and Novak (1987) describe a derivation
of all the complex stiffness constants for flexible caps, rigid
caps, and piles with separation (gapping).
4.2.5.3 Battered piles—Pile batter can be considered by
calculating the pile stiffnesses for a vertical pile, assembling
the stiffnesses as the stiffness matrix [K] in element coordinates (along and perpendicular to the axis of the pile), and
transforming this matrix into horizontal and vertical global
coordinates (Novak 1979). This gives the pile stiffness matrix
[k′]j = [T]T[k]j[T]
cos α sin α 0
[ T ] = – sin α cos α 0
0
0 1
(4-38)
where α = the angle between the battered pile and vertical.
The pile stiffness matrix in global coordinates becomes
k u′
k uv
′ k u′ ψ
′
k vu
k ′v k v′ ψ
=
(4-39)
′ k′ψv k ψ′
k ψu
2
2
k u cos ( α ) + k v sin ( α ) cos ( α ) sin ( α ) ( k u – k v ) cos ( α )k u ψ
2
α ij∗ = α ij ( 1 ) + i ⋅ α ij ( 2 ) =
(4-37)
in which the transformation matrix [T] depends on direction
cosines from the batter. The horizontal and vertical stiffnesses for the individual battered pile can then be combined
with the other pile stiffnesses as presented in Section 4.2.5.2
for pile groups.
When the horizontal coordinate axis lies in the plane of the
batter, the transformation matrix is
∑r cr . Better estimates
may be obtained using dynamic interaction coefficients.
Evaluation of group effects using dynamic interaction
coefficients—The dynamic interaction factor αij is a dimensionless, frequency-dependent, complex number
351.3R-41
2
cos ( α ) sin ( α ) ( k u – k v ) k u sin ( α ) + k v cos ( α ) sin ( α )k uψ
(4-36)
cos ( α )k ψu
sin ( α )k ψu
kψ
351.3R-42
ACI COMMITTEE REPORT
determined with respect to the center of gravity. Fig. 4.11 is
presented for a simple system where both embedment and
bottom support are provided. Evaluation of the forces
associated with free-body movements yields an impedance
matrix for the system with respect to the center of gravity of
K∗uu
K∗u ψ
K∗u ψ K∗ψψ
k∗u + k∗eu
=
(4-41)
– ( k∗u y c + k∗eu y e )
2
2
– ( k∗u y c + k∗eu y e ) k∗ψ + k∗u y c + k∗eu y e
where
Kij* =
Fig. 4.11—Coupling effect introduced by an elevated CG.
The element stiffness functions k are calculated assuming
that the pile is vertical, thus
kuv = kψv = kvψ = 0 and kuψ = kψu
(4-40)
In some cases, the off-diagonal terms of the transformed
stiffness matrix may be ignored and only the diagonal terms
carried forward. One criterion for this is if the off-diagonal
terms are small in comparison to the diagonal terms (for
example, less than 10% on an absolute value comparison).
The same process may be applied to the damping terms, or,
for more accuracy, the transformation can take place using
the complex impedance functions.
4.2.6 Transformed impedance relative to center of gravity—
A machine foundation implies a body of certain depth, and,
typically, the center of gravity of the system is above the center
of resistance provided by the soil, embedment, piles, and any
combination of these. For simplicity of analysis, many foundations are treated as a rigid body, and their center of gravity is
used as the point of reference for all displacements and
rotations. Thus, stiffness or impedance provided by the support
system (piles, isolation springs, soil, or soil embedment) has to
be transformed to reflect resistance provided against motions
of the center of gravity. In this analysis, a horizontal translation
is resisted not only by horizontal soil reactions but also by
moments. This gives rise to a coupling between translation and
rotation and the corresponding “off-diagonal” or cross stiffness
and damping constants such as kuψ = kψu and cuψ = cψu.
Double subscripts are used to indicate coupling. (Note that
Eq. (4-25d) and (4-25e) also introduced coupling of terms.)
For coupled horizontal and rocking motions, the generation
of the stiffness and damping constants is developed as shown
in Fig. 4.11. By applying unit translations to a free body of the
foundation, examining the forces developed in the support
system by the translation, and determining the forces needed
to cause this unit translation, the coupled impedances can be
impedance in the i-th direction due to a displacement in the j-th direction;
= distance from the CG to the base support;
yc
ye
= distance from the CG to the level of embedment
resistance; and
other terms are as previously defined.
If the CG is not directly over the center of vertical impedance,
additional coupling terms between the rotation and the vertical
motion are introduced. Thus, most engineers diligently adhere
to guidelines to minimize such in-plan eccentricities. In
extreme cases, it may be appropriate to develop the K*
matrix as a full six-by-six matrix due to eccentricities. For
other combinations of directions in other coordinate systems,
the sign on the off-diagonal terms may change, so proper
diligence to sign convention is required. This transformation
to the CG may be developed on stiffness and damping terms
or based on impedance.
4.3—Vibration analysis
4.3.1 Foundation stiffness—For a concrete block foundation
on grade, when the foundation block is thick enough, the
foundation block can be considered rigid; a formula used for
this judgement is
T > 2 + LB /30 (SI: T > 0.6 + LB /30)
(4-42)
where T is the foundation thickness in ft (m), and LB is the
greater plan dimension of the foundation block in ft (m).
Otherwise, the foundation block is considered flexible in the
vibration analysis.
Equation (4-42) is based on engineering judgement and
the industry practice for some traditional block-type foundations (Fig. 2.4). The applicability of Eq. (4-42) to large
combined block foundations (Fig. 2.5) is not well established
and may need to be investigated by the engineer on a caseby-case basis. Finite element dynamic analysis shows that
some of these large combined block foundations may not
behave as rigid foundations. Analysis of the frequency and
dynamic response of such foundations using the finite
element methods given in Section 4.3.5 may be appropriate.
FOUNDATIONS FOR DYNAMIC EQUIPMENT
In many applications, concrete provides the required inertial
mass so that foundations are often massive and rigid. This, in
conjunction with the rigid nature of many machines, can
greatly simplify dynamic analysis of the machine-foundation
system. For equipment on a rigid foundation, when the position
of all parts of the system can be described by a single variable
at any time, the system can be accurately represented by a
SDOF model. For example, pure vertical or torsional motion of
a rigid system can be accurately represented by a SDOF model.
Nevertheless, in most rigid machine-foundation systems,
the horizontal motion is coupled with the rocking motion due
to the system center of gravity (CG) being at one height and
the center of resistance (CR) being at a different level. A two
DOF model represents the coupled response of the system
for this condition. In general, a six DOF (degree of freedom)
model is needed to properly represent the dynamic performance of a rigid machine-foundation system. If the foundation
is well laid out so that the CG and the CR are positioned over
each other (vertically aligned), then the six DOF model
mathematically uncouples to become two problems of two
DOF (rocking about one horizontal axis coupled with translation along the other horizontal axis for both horizontal
directions) and two SDOF problems (vertical motion and
torsional motion about the vertical axis).
4.3.2 Single degree-of-freedom system—For a SDOF
system, a closed form solution yields the fundamental
natural frequency as
1
f n = ---------- k ⁄ m
2⋅π
(4-43)
where
fn = system natural frequency in cycles/second;
k = the dynamic stiffness provided by the supporting
media obtained in accordance with Section 4.2, the
spring constant of the system; and
m = mass of the machine-foundation system.
If the particular SDOF system under investigation
involved rotational motions, the stiffness term in Eq. (4-43)
would be the applicable rocking or torsional stiffness and the
mass term would be a mass moment of inertia. Thus, the
stiffness and mass terms are specifically associated with a
specific direction of motion of the machine-foundation
system. The calculated natural frequency is then compared
with operating frequencies of the machine to ensure that the
frequency criteria as set forth in Section 3.4 are satisfied.
A forced response analysis is performed to determine
amplitudes of vibration. The forced response analysis can be
a harmonic analysis if the forcing function is harmonic. A
closed form solution for a harmonically excited SDOF
system yields
Fo ⁄ k
A = --------------------------------------------------------------------------------2 2
2
( 1 – ( ω o ⁄ ω n ) ) + ( 2βω o ⁄ ω n )
where
(4-44)
351.3R-43
A = displacement amplitude;
Fo = dynamic force amplitude;
k = the dynamic stiffness provided by the supporting
media;
ωo = circular operating frequency of the machine (rad/s);
ωn = circular natural frequency of the machine-foundation
system = (k/m)1/2; and
β = system damping ratio = ci /[2(k/m)1/2].
A SDOF analysis can be a very effective tool in designing
a rigid foundation. When used in conjunction with a parametric study the use of a SDOF in the direction of interest
can often bound the solution. Furthermore, the use of an
approximate SDOF analysis can show the feasibility of satisfying particular vibration criteria and can provide a rough
check on more-detailed analysis.
4.3.3 Two degree-of-freedom system—As mentioned
earlier, a SDOF system may not be sufficient to represent a
rigid machine-foundation system. For example, a dynamic
force from a rotating machine acting on shaft supports causes
translation and rocking motions of the system, and a two DOF
system is a more accurate representation of the system. For a
two DOF system, the following frequency equation can be
used in solving for the two natural frequencies
2
4
ωn
K uu K ψψ
K uu ⋅ K ψψ – K uψ
2
– ------- + ---------- ⋅ ω n + -------------------------------------- = 0
m
Iψ
m ⋅ Iψ
(4-45)
where
ωn = natural frequencies of the system;
Iψ
= mass moment of inertia of the system about the CG;
m
= mass of the machine-foundation system;
Kuu = horizontal spring constant Eq. (4-41);
Kuψ = coupling spring constant Eq. (4-41); and
Kψψ = rocking spring constant Eq. (4-41).
Amplitudes of motion for the two DOF can be calculated
in a manner similar to Eq. (4-44) as a modal combination.
When rotational (rocking or torsional) motions of the foundation exist, it is important to compute displacements at the
key points of interest machine bearings, for example, by
combining the rotational effects with the translational
motions. These combined motions may be less than or
greater than the motions of the CG, which are determined
through the modal combination.
4.3.4 Three or more degree-of-freedom system—Except
for cases described in Sections 4.3.2 and 4.3.3, machines
supported on flexible or even rigid foundations generally can
not be adequately represented by a single or two DOF model,
and a three or more DOF system should be used.
4.3.4.1 Mathematical models—The physical structure can
be simplified and represented by a mathematical model for
dynamic analysis. The model should include the machine,
structure, and stiffnesses of supporting medium (soil or piles).
The current state-of-the-art in machine foundation design is
limited to linearly elastic theory; nonlinear behavior generally
is not considered in routine foundation design.
4.3.4.2 Frequency analysis—A frequency analysis is
performed on the mathematical model to calculate natural
351.3R-44
ACI COMMITTEE REPORT
frequencies and mode shapes of the system. This calculation
is also called normal mode or free vibration analysis. The
general dynamic equation of motion is
[ M ]X·· ( t ) + [ C ]X· ( t ) + [ K ]X ( t ) = F ( t )
(4-46)
where
[M] = mass matrix;
[C] = damping matrix;
[K] = stiffness matrix; and
F(t) = time varying force vector.
In Eq. (4-46), X and its derivatives represent displacement,
velocity, and acceleration, respectively, of various points on
the machine-foundation system.
For a frequency analysis, the force and damping terms are
set to zero. The equation of motion reduces to:
[ M ]X·· ( t ) + [ K ]X ( t ) = 0
(4-47)
Using the normal mode substitutions (common to all
classical vibration analysis such as in: Arya, O’Neill, and
Pincus 1979; Harris 1996), the following equation is developed
2
( [ k ] – ωi [ m ] ) ⋅ Ψi = 0
(4-48)
where
[k] = reduced stiffness matrix;
[m] = reduced mass matrix;
ωi
= circular natural frequency for the i-th mode; and
Ψi = reduced mode shape vector for the i-th mode.
Solving Eq. (4-48), the natural frequencies (eigenvalues)
and mode shapes (eigenvectors) are obtained. An examination
is then made to determine how the frequencies associated
with the excitation forces generated by machine operation
match up with the natural frequencies of the system. If these
operating frequencies are close to the computed system
natural frequencies, a resonance condition exists. Section
3.4.3 discusses the criteria for frequency separation.
4.3.4.3 Forced response analysis—In the forced
response analysis, Eq. (4-46) is solved for X, the displacements of the machine-foundation system as a function of
time. The forced response analysis can take the form of a
harmonic frequency analysis or a time history analysis. The
selection of method depends on the forcing functions to be
evaluated. For the analysis of rotating or reciprocating
machinery foundations, the harmonic analysis method is
used extensively to determine the steady-state response of a
linear structure to a set of given harmonic loads.
Harmonic analysis can be carried out either by direct solution
of the equations of motion or by mode superposition techniques. In the direct solution method, the equations of motion
can be solved in the time domain. This method does not require
a natural frequency analysis (Eq. (4-47) and (4-48)). This
numerical method is more general and flexible than the
mode superposition method. In the mode superposition
method, a natural frequency analysis is performed first. The
forced response analysis is then carried out on a mode-bymode basis, in some cases for a limited number of modes.
These modal results are summed to obtain forced response
analysis results. The summation may consider the phase
relationship between the modes or may conservatively
consider only the amplitudes of the modal responses
depending on the level of accuracy desired.
The results of the forced response analysis include
displacements, velocities, and accelerations. These results
are then compared against allowable limits for acceptance.
The allowable limits are typically applicable at discrete
locations (for example, the equipment bearing, connections
to adjacent equipment, and locations where people may be
affected). For these multi-DOF models representing structural
and machine flexibility, the modeling commonly includes
node representations for the locations of interest. The analysis
should account for variation in parameters such as structural
stiffness, mass, and soil properties.
4.3.5 Dynamic analysis using computer codes—Although
it is possible to calculate frequencies and responses of a
machine-foundation system using equations, practicing
engineers find it is often impractical and inefficient to do so,
especially for flexible systems with multiple DOF. Many
commercially available computer codes, such as ANSYS,
DYNA, GTSTRUDL, RISA, SACS, SAP2000, and
STAAD, can be used to perform these tasks.
In the computer application, structural beams and columns
are generally represented by prismatic elastic members, and
concrete floors and mats are represented by plates or solid
elements. Stiffnesses of the foundation soil or piles should be
properly included in the model to take into account the soilstructure interaction effects. Section 4.2 discusses the
impedance of the supporting media. The machine can be
quite stiff compared with the structural members. If so, the
machine can be represented by a mass point or points
concentrated at their centers of gravity and connected to their
supports using rigid links. Solid elements with matching
densities can also be used to simulate the mass and mass
moment of inertia of the machine. If the equipment is not so
stiff as compared with the structural members, as is the case
for many large gas compressors, its flexibility should be
included in the computer model representation.
4.4—Structural foundation design and materials
4.4.1 Reinforced concrete—The basic principles of
reinforced concrete design embodied in ACI 318 are used in
the design of foundations supporting dynamic equipment.
Engineers working with specific equipment may apply
additional criteria based on their experience. Such equipment
includes forging hammers, turbine-generator systems, and
large compressors.
The general equipment foundation includes some components that primarily act in flexure, others that are primarily
axial, and others may act primarily in shear. Applicable
sections of ACI 318 are often used to establish minimum
requirements for axial, flexural, and shear reinforcement. In
some cases, engineering firms have supplemented these
criteria with internal criteria or internal interpretations of
FOUNDATIONS FOR DYNAMIC EQUIPMENT
ACI 318 requirements. Such steps are taken for foundation
structures that are very different from conventional building
structures (for example, turbine-generator pedestals). For
thick sections, special criteria involving location of reinforcing
or minimum reinforcing may be identified more in line with
mass concrete construction. Such criteria are typically structure
specific (for example, only for turbine-generator pedestals or
only for foundation slabs over 6 ft (1.8 m) thick) and, thus,
are not extendable to the broad class of foundations
addressed in this document.
Largely because of the broad range represented in this
class of construction, accepted standards have not evolved.
For example, there is no specific minimum amount of
reinforcement applicable across the board for these designs.
In some applications, building code requirements may apply
and machine manufacturers may set minimum standards. A
minimum concrete strength of 3000 psi (21 MPa) can be
applied. For most foundations and foundations supporting
API equipment, a strength of 4000 psi (28 MPa) is
commonly specified and may be required. In most cases, the
quality and durability of the concrete is considered more critical
to good performance than strength.
As with most construction, ASTM A 615 Grade 60
reinforcing steel is most commonly used for dynamic
machine foundations. Good design practice requires particular
attention to the detailing of the reinforcement, including
proper development of the bars well into the block of the
concrete, avoidance of bar ends in high stress regions, and
appropriate cover. Excessive reinforcement can create
constructibility and quality problems and should be avoided.
Some firms specify a minimum reinforcing of 3.1 lbf/ft3
(50 kg/m3 or 0.64%) for piers (machine support pedestals)
and 1.9 lbf/ft3 (30 kg/m3 or 0.38%) for foundation slabs. For
compressor blocks, some firms suggest 1% reinforcing by
volume and may post-tension the block (Section 4.4.1.5).
Many engineers recommend additional reinforcement, such
as hairpin bars, around embedded anchor bolts to ensure
long-term performance. The criteria and presentations in
ACI 351.2R for static equipment also can be applied to
dynamic equipment foundations.
4.4.1.1 Fatigue issues—For many dynamic equipment
foundations, the cyclic stresses are small, and engineers
choose to not perform any specific fatigue stress calculations.
Other equipment can require more significant consideration
of cyclic stresses. In such cases, ACI 215 provides guidance,
particularly where the flexural characteristics of the foundation are most important.
Some of the methods used by firms to implicitly or explicitly
address fatigue include:
• Proportioning sections to resist all conventional loads
plus three times the dynamic load;
• Designing such that concrete modulus of rupture is not
exceeded while including the inertial loads from the
concrete motion. In certain cases, the computed modulus
of rupture is reduced by 50% to approximate permissible
stresses reduced for fatigue;
• Reducing by as much as 80% the strength reduction
factors specified by ACI 318; and
351.3R-45
•
Recognizing that cracking is less likely in structures
built with clean, straight lines and not having re-entrant
corners and notches.
4.4.1.2 Dynamic modulus of elasticity—The dynamic
modulus of elasticity is stiffer than the static modulus,
although not in any simple form. Established relationships
suggest that the ratio of dynamic to static modulus can vary
from 1.1 to 1.6, with significant variation with age and
strength. In practice, engineers treat this strain-rate effect
differently. In some firms, engineers perform calculations
using the higher dynamic modulus while other firms and
engineers consider the difference unimportant and use the
static modulus from ACI 318. The distinction is more important for elevated tabletop-type foundations where the frame
action of the structure is stiffer if a dynamic modulus is used.
The difference can also be important in compressor foundations where the stiffness of the machine frame must be evaluated against the stiffness of the concrete structure. For
simple, block-type foundations, the concrete modulus of
elasticity has no real effect on the design.
4.4.1.3 Forging hammer foundations—To the committee’s
knowledge, there is no current U.S. document addressing
design requirements for forging hammer foundations. Most
hammer manufacturers are familiar with German DIN
Standard 4025. That document is summarized in the
following paragraphs for general information.
The required weight of a foundation block sitting on soil
should be determined by calculation, and such calculations
should consider the effect of vibrations on nearby structures and
facilities. One reference equation suggested by DIN 4025 is
v 2
W f = 75 ⋅ B r ⋅  ----r 
 v o
where
Wf =
Br =
vr =
vo =
(4-49)
weight of the foundation, tons (kN);
ram weight, tons (kN);
ram impact velocity, ft/s (m/s); and
reference velocity 18.4 ft/s (5.6 m/s) from a free fall
of 5.25 ft (1.6 m).
This equation assumes an anvil-to-ram weight ratio of
20:1. The foundation weight should be increased or
decreased to make up for a lighter or heavier anvil.
The design of the foundation block considers a statically
equivalent load determined from the impact energy and
characteristics of the forging process in addition to other
design basis loads. Minimum reinforcement of the foundation
block is set at 1.56 lbf/ft3 (25 kg/m3) or 0.32%. This reinforcement should be distributed in all three directions
throughout the block. The upper layer of steel should be
capable of resisting 1% of the statically equivalent load
applied in any horizontal direction. Bending and shear
effects should be addressed in the layout and design of the
reinforcement. In large hammer foundations, reinforcement
is often installed in all three orthogonal directions and
diagonally in the horizontal and vertical planes. Suitable
development of the reinforcement is very important.
351.3R-46
ACI COMMITTEE REPORT
Fig. 4.12—Types of machinery frame support systems
(courtesy of Robert L. Rowan & Associates).
4.4.1.4 Thermal effects—Some types of dynamic equipment also develop greater than normal thermal conditions, with
concrete surface temperatures exceeding 150 °F (66 °C)
around and within the foundation. This is especially true for
combustion turbines, steam turbines, and compressors. The
engineer should address the effects of these thermal conditions in the design phase. Inadequate consideration of the
thermal effects can lead to early cracking of the foundation,
which is then further increased by the dynamic effects.
Calculation of thermal induced bending requires proper
determination of the heat distribution through the thickness
of the foundation. ACI 307 provides some guidance that can
be extrapolated to hot equipment. ACI 349.1R also provides
methodologies that may be transferable to certain machine
foundations. Heat transfer calculations can also be
performed either one-, two-, or three-dimensionally.
The most effective methods of controlling the thermal
effects are:
• Provide sufficient insulation between the hot equipment
and the concrete;
• Provide sufficient cover to the reinforcement so that
thermally induced cracking neither degrades the bond
of the reinforcement nor increases the exposure of the
reinforcement to corrosives; and
• Provide sufficient reinforcement to control the growth
of thermal induced cracks.
4.4.1.5 Compressor block post-tensioning—Some
engineers prefer that block-type compressor foundations be
post-tensioned to provide residual compressive stress that
will prevent the generation of cracks. Shrinkage cracks or
surface drying cracks are expected in any concrete block
foundation, especially when the water content is excessive.
With the addition of subsequent vibration, these cracks propagate, allowing oil to penetrate the block, and eventually
destroy the integrity of the foundation. Post-tensioning puts
the block in compression, offsetting the dynamic and
shrinkage stresses. When horizontal post-tensioning rods are
placed 1/3 the distance from the top of the block, a triangular
compression stress distribution can be idealized. This idealization maximizes the compression at the top where it is
needed the most. An average pressure of 100 psi (690 kPa)
translates to 200 psi (1380 kPa) at the top, providing the
necessary residual compression.
Vertical post-tensioning rods are anchored as deep as
possible into the foundation mat and are sleeved or taped
along their length to allow them to stretch. The embedded
end is anchored by a nut with a diameter twice the rod diameter and a thickness 1.5 times the rod diameter. Horizontal
rods are nonbonded (sleeved) and anchored at each end of
the block through thick bearing plates designed to distribute
the load on the concrete. High-strength steel rods are recommended for post-tensioning compressor blocks (Smalley and
Pantermuehl 1997).
The concrete should have a 28-day strength of at least
3500 psi (24 MPa), superior tensile strength, and, when
cured, be reasonably free of shrinkage cracks. Compliance
with ACI 318 Sections 5.8 to 5.13 (Mixing, Conveying,
Depositing, Curing, and Hot and Cold Weather Requirements) is commonly mandated for these systems.
4.4.2 Machine anchorage—The major components of
machine anchorage are the anchor bolts and the support
system directly under the machine frame at the anchor bolt
location. Support systems range from a full bed of grout to
various designs of soleplates and chocks, fixed or adjustable,
shown in Fig. 4.12. Additionally, isolation support systems
are discussed in Section 4.5. Various styles of anchor bolts
are shown in Fig. 4.13.
4.4.2.1 Performance criteria/anchor bolts—The structural
performance criteria for anchor bolts holding dynamic
machinery require that sufficient clamping force be available
to maintain the critical alignment of the machine. The
clamping force should allow smooth transmission of unbalanced machine forces into the foundation so that the machine
and foundation can act as an integrated structure. Generally,
higher clamping forces are preferred because high clamping
forces result in less vibration being reflected back into the
machine. In the presence of unbalanced forces, a machine
that has a low clamping force (400 psi [2.8 MPa]) at the
machine support points can vibrate more than the same
machine with high clamping forces (1000 psi [7 MPa]).
Precision machines in the machine tool industry sometimes
have clamping forces as high as 2000 psi (14 MPa) to minimize
“tool chatter.” Instead of more refined data, designing for a
clamping force that is 150% of the anticipated normal
operating bolt force is good practice. A minimum anchor
bolt clamping force of 15% of the bolt material yield strength
is often used if specific values are not provided by the equipment manufacturer. Higher values are appropriate for more
aggressive machines. Clamping force, also known as
preload, is developed by pretensioning the anchor bolt.
FOUNDATIONS FOR DYNAMIC EQUIPMENT
351.3R-47
Table 4.3—Anchor bolt materials—mechanical
properties—inch products
ASTM
designation
Grade
A 36*
—
A
A 307†
A
193*
C
B7
Yield
Tensile
strength, min.,
Diameter, in. strength, min. 0.2% offset,
(mm)
ksi (MPa)
ksi (MPa)
—
50 to 80
(400 to 550)
36 (250)
4 (102) and
smaller
60 (414)
58 to 80
(400 to 550)
—
36 (250)
2-1/2 (64) and
smaller
over 2-1/2 (64)
to 4 (102)
125 (860)
115 (795)
105 (720)
95 (655)
58 to 80
(400 to 552)
36 (248)
75 to 95
(517 to 655)
125 to 150
(862 to 1034)
55 (380)
36
F 1554†
Fig. 4.13—Evolution of anchor bolt design (courtesy of
Robert L. Rowan & Associates). Note: J- and L-bolts are
generally not used for dynamic equipment foundations or in
highly stressed applications.
4.4.2.2 Capacity—The capacity of each anchor bolt should
be greater than design loads to provide adequate reserve
capacity. Conditions can change over time due to machine wear
or changes in operating conditions. Properties as given in the
cited ASTM standard specifications of the steels commonly
used for anchor bolts are listed in Table 4.3 and 4.4. Because
the number and diameter of anchor bolts are determined by
the machine manufacturer, the engineer can maximize
capacity by specifying the higher-strength steels. The practicable capacity of an anchor bolt is typically 80% of the yield
strength, not the full tensile strength.
4.4.2.3 Anchor bolt preload—To avoid slippage under
dynamic loads at any interface between the frame and chock
and soleplate, or chock and foundation top surface, the normal
force at the interface multiplied by the effective coefficient of
friction must exceed the maximum horizontal dynamic force
applied by the frame at the location of the tie-down.
In general, this requires
Fr = µ(Tmin + Wa) or Tmin = Fr /µ – Wa
(4-50)
where
Fr = maximum horizontal dynamic force;
µ
= coefficient of friction;
Tmin = minimum required anchor bolt tension; and
Wa = equipment weight at anchorage location.
An anchor bolt and concrete anchorage system that has
long-term tensile strength in excess of Tmin and maintains
preload at or above this tension, coupled with a chock interface
whose coefficient of friction equals or exceeds µ, will withstand
the force Fr to be resisted. A conservative approach neglects
Wa (assumes it to be zero) because distortion of the frame or
block may reduce the effective force due to weight at any one
anchorage location.
In the case of machines, such as reciprocating gas
compressors, gas loads or inertia loads may dictate the
55
4 (102) and
smaller
105
105 (724)
*
The values stated in inch-pound units or SI units are to be regarded separately as standard; each system must be used independently of the other. Do not combine values
from the two systems.
†
The values stated in inch-pound units are to be regarded as the standard.
Table 4.4—Anchor bolt materials—mechanical
properties—metric products
ASTM
designation
A 36M
A 193M
Grade
—
B7
Tensile
strength,
min. MPa
Diameter, mm
—
400 to 550
M64 and
smaller
M64 to M100
Yield strength
min., 0.2%
offset, MPa
250
860
720
795
655
required frictional holding capacity Fr (Section 3.2.3;
Smalley and Pantermuehl [1997]), depending on the location
of the anchor bolt. Because holes in the frame and cross-head
guide establish bolt diameter, the bolt material yield strength
determines the maximum possible preload.
The Gas Machinery Research Council (GMRC) research
program has set out to develop data for friction between
common chock interface materials, including steel/cast iron,
steel/steel, epoxy/cast iron, epoxy/steel, epoxy/epoxy, dry,
and with oil present, using sizes that come close to those
typical of compressor mounting practice (Smalley 1997). This
report presents some values for “breakaway” friction coefficients including a range from 0.22 to 0.41 for dry interface
between cast iron and various epoxy products and a value of
0.19 for cast iron on cold rolled steel. The presence of oil in
the sliding interfaces reduces the friction coefficient for cast
iron on epoxy to a range from 0.09 to 0.15 and to a value of
0.14 for cast iron on cold rolled steel. Thus, maintaining an oil
free interface greatly enhances frictional holding capacity.
Example:
A separate analysis has shown that each 2 in. (50 mm)
diameter anchor bolt of A 193 Grade B7 material for a
compressor should carry a maximum horizontal dynamic
load of 13,500 lbf (60 kN). What preload tension should be
maintained in the anchor bolt?
Using a coefficient of friction of 0.12 and setting the
contribution of compressor weight to zero, Eq. (4-50)
gives the following minimum tension in the anchor bolt
351.3R-48
ACI COMMITTEE REPORT
Tmin = 13,500 lbf /0.12 = 112,500 lbf = 500 kN.
The recommended clamping force (lacking more explicit
information) is 150% of this value or 168,750 lbf (750 kN).
The nominal area of a 2 in.-diameter bolt is 3.14 in.2
(20.3 cm2). With a yield stress of 105,000 psi (724 MPa),
the yield force for the bolt is 330,000 lbf (1467 kN). The
required force is 51% of the yield force, which is less than
the 80% maximum and greater than the 15% minimum.
The bolt should be preloaded to 168,750 lbf (750 kN) and
that preload maintained. To compute a minimum required
yield stress for this application
168,750 lbf
Bolt stress = --------------------------- = 53,700 psi (375 MPa)
2
3.14 in.
53,700 psi
Required Fy = ------------------------- 67,200 psi (463 MPa)
80%
A material with a yield stress in excess of 67,200 psi (463
MPa) could be substituted for the A 193 Grade B7 material.
4.4.2.4 Monitoring preload—Anchor bolts always lose a
portion of the preload both in the first 24 h after tightening
and then during operation. Usually, at least one retightening
is required until the preload stabilizes. Periodic retightening
to the original preload level may be required. Bickford
(1995) and field observations explain the reasons for this
retightening. Some types of machines may require shorter
intervals between retightening (after the initial series), but
common periods are 6 months to 1 year.
To aid in knowing if the preload loss is excessive, there are
electrical and mechanical methods of measuring the preload
while the machine is operating. Most electrical methods
involve strain gages and read-out devices. Their application
is limited to almost laboratory-type conditions. Mechanical
methods incorporate the equivalent of a depth micrometer that
measures the increase in length of the top of the bolt over a 3 to
4 in. (75 to 100 mm) distance. The UK-patented RotaBolt™
uses the aforementioned principle with a permanent pin and
cap in the top of the bolt and achieves an accuracy of 5%.
Other products with related mechanisms exist in the marketplace. With any such device, the engineer should seek data
and analysis to ensure its fitness for the planned application.
There also have been attempts to incorporate both electrical
and mechanical methods into “force” washers, but no extensive service history is available at this time. Ultrasonic
methods can accurately measure the increase in length of a
cleanly terminated bolt (not a J or L) and are available from
service providers.
4.4.2.5 Depth/length/style—Figure 4.13 shows the
various styles of common anchor bolt designs. A good practice is to make the anchor bolt as long as possible so the
anchoring forces are distributed lower in the foundation or
ideally into the concrete mat below the foundation pier.
Anchor bolts that are designed to exactly match a ductile
failure criterion, just long enough so that the concrete pullout
strength equals the strength of the steel, are too shallow for
dynamic machinery foundations. Cracking of the upper
concrete has been a common problem when shallow
embedment depths are used.
There are additional benefits to using a long anchor bolt.
Such systems exhibit greater tolerance to grout creep (that is,
less loss of preload from creep). In addition, the lower termination point, in the foundation or in the mat below, moves
the potential site for any crack initiation away from the
dynamic loads imposed by the machine and away from
sources of oil.
In addition to the depth, the engineer should pay attention
to the bolt style. J- and L-style bolts can straighten out and
pull out of concrete foundations before their maximum
tensile capacity is reached. Many engineers prohibit their use
with dynamic machinery. Expansion shell anchors should
only be used where they have been tested by the manufacturer and approved for the vibratory condition of the particular dynamic machinery application.
4.4.3 Grout—The grout chosen should provide the longterm strength to carry the applied load from the machine
mounts. For dynamic equipment foundations, engineers often
specify cementitious grout (also called hydraulic machine
base grout) or polymer grout (epoxy machine base grout is one
popular type of polymer grout). For dynamic machines,
polymer grouts are often specified because of better resistance
to vibration and impact loads and better chemical resistance to
process fluids and lubricating oils. The engineer, however,
should consider cementitious grouts where they can meet the
requirements because they cost less. Cementitious grouts can
have compressive strength as high as 6000 psi (42 MPa) but
are low in tensile and flexural strength, thereby limiting their
use for dynamic machines to smooth running rotating
machines such as electric generators.
Polymer grouts can be formulated much stronger in
compression than cementitious grouts, with strengths up to
16,000 psi (110 MPa) possible. With machine loads generally
less than 1000 psi (7 MPa), high compressive strengths
should not be the selection criterion. The higher compressive
strength polymer grouts tend to be more brittle and crack
readily. Additionally, the engineer should consider other properties such as the modulus of elasticity and creep at operating
temperature. Creep, the tendency of materials to exhibit timedependent deformation, can be a problem, especially with
polymer grouts. Their deflection under load increases with
time; higher temperatures and thicker grout layers aggravate
the tendency. In general, low creep and high modulus of
elasticity are desirable. There are products in the marketplace
with projected 10-year creep at 140 °F (60 °C) under 600 psi
(4.2 MPa) of 0.004 in./in. (mm/mm) or less (tested by
ASTM C 1181), and short-term (1 h) modulus of elasticity
close to or above 2 × 106 psi (13,800 MPa) at 125 °F (52 °C)
(ASTM C 580). Creep generally increases as the modulus of
elasticity decreases, and both are brand-specific properties.
Users are cautioned that increasing grout depth beyond
normal thickness of 2 to 4 in. (50 to 100 mm) increases the
deformation due to creep and elastic shortening.
Polymer grouts can exhibit slight shrinkage on curing, but
such shrinkage is not detrimental to operating life, as
FOUNDATIONS FOR DYNAMIC EQUIPMENT
evidenced by over 40 years of use of such products. Most
polymer grouts require expansion joints on 3 to 4 ft (900 to
1200 mm) centers because the coefficient of thermal expansion
of polymer grouts is greater than concrete and because of the
exothermic reaction of the grout during curing. Typical
thicknesses for grout are 2 to 4 in. (50 to 100 mm); however,
the maximum allowable thickness can be formula specific.
Some formulations permit thicknesses up to 18 in. The engineer
should assess any mounting system under load for long-term
performance considering deflection, creep, applied loads,
grout thickness, and operating temperature, relative to the
performance criteria in Sections 3.5 and 3.6.
ACI 351.1R provides a more complete explanation of the
properties of both polymer and cementitious grouts. Several
reports by the GMRC address friction properties, creep properties, and methods of engineering and assessing epoxy
grouts and chock material for reciprocating compressor
applications (Smalley and Pantermuehl 1997; Smalley 1997;
Pantermuehl and Smalley 1997a,b).
4.5—Use of isolation systems
Isolation systems create a dynamic system with stable,
identifiable properties. The consistency of the isolator
properties allows other conservative factors to be eliminated. The isolation system provides the stiffness and
damping to the system with the equipment and inertia block,
if applicable, providing the mass. Most commonly the stiffness
provided to the machine-foundation system by an isolation
system will be less than that provided by the soil or piles.
The isolation system allows the machine-foundation mass to
“float” to reduce the transmission of vibrations. The inertia
(mass) of the system should resist the dynamic forces from
the equipment. The growing practice of base-isolation for
the seismic design of buildings has its roots in isolation of
machinery vibration.
There are three basic isolator concepts generally used for
dynamic machinery applications: rubber pad-type materials,
steel springs often combined with viscous dampers, and air
mounts. Rubber pad-type materials are generally the stiffest
designs and have a nonlinear load deflection relationship.
Steel spring systems are softer, providing better isolation.
Whereas the springs themselves lack significant damping,
these designs can be augmented with viscous dampers,
providing damping levels from 5 to 40%, depending on the
particular application needs. Air-mount systems are the
softest type isolator.
4.5.1 Direct support systems—With direct isolator support
of the equipment, the equipment and isolators form the
complete dynamic system. The foundation supporting the
isolators is subject only to static loads, with the isolators
greatly reducing the dynamic force component.
The isolator manufacturer or engineer should arrange the
isolators under the equipment in a manner that properly
stabilizes the equipment, distributing both stiffness and
damping. When isolator manufacturers perform this task,
they should provide the foundation engineer with all the
necessary design loads for the foundation. If sufficient information is provided (mass distribution, dynamic loads, and
351.3R-49
Fig. 4.14—Foundation deterioration caused by cracking.
performance requirements), the isolator manufacturer can
often evaluate the motion of the equipment against the established limitations. Essentially, these calculations are the
same as those made for soil or pile-supported systems with
the stiffness and damping coming from the isolators.
4.5.2 Inertia block systems—Inertia block systems are
used where the mass of the equipment is insufficient to limit
the vibration response of the system to acceptable levels. For
both direct support systems and inertia block systems, the
same basic principles in laying out the isolators apply.
Likewise, the same basic calculations for the dynamic motions
apply. The design of the inertia block follows concrete design
requirements established elsewhere in this document.
4.6—Repairing and upgrading foundations
4.6.1 Introduction—Concrete foundations may need
repairing after time because of greater-than-anticipated
loads, use beyond the design life, inadequate original design,
or inappropriate maintenance. Cracking and deterioration of
the concrete that affects the machine’s performance are indicators for repair. Through field experience, cracks larger
than 0.016 in. (0.4 mm) have been found to be wide enough
to allow penetration by fluids, such as oil, which can thereby
can cause a crack to grow due to hydraulic action. Furthermore, cracks can cause anchor bolts to loosen and start a
vicious cycle (foundation deterioration causes machine
deterioration that, in turn, increases foundation deterioration) as
illustrated in Fig. 4.14. Because concrete always cracks, it is
important to evaluate whether the cracks present a cosmetic
problem or a problem in machine performance.
4.6.2 Upgrading—When the engineer considers repairing
the foundation, he or she may also consider upgrading the
foundation. If the repair is needed because the original design
was inadequate, the loads increased, the machine was
upgraded, or the original technology is significantly below
current standards, then incorporating upgrades into the repair
design makes sense. It is illogical to repair a foundation back
to its original condition after operation has proven the original
design to be inadequate. Application of current technology
will likely provide a longer life of trouble-free service than has
been experienced with the foundation in the past.
4.6.3 Accomplishing repairs and upgrades—Typically, the
condition of the concrete in foundations needing repair is worse
in the upper quarter of the foundation where loads are greater
and deterioration from process fluids or lubricating oil are more
351.3R-50
ACI COMMITTEE REPORT
Fig. 4.15—Cross section of a foundation repair (courtesy of Robert L. Rowan & Associates).
likely to be present. Often, a foundation in need of repair can be
restored by removal of the top 18 to 24 in. (450 to 600 mm) of
old concrete and grout. In this relatively thin upper section, the
construction crew can install a strong “bridge” of repair material
that can withstand the machine loads and distribute them
uniformly to the structure left in place below.
Typical elements of an engineered design for upgrading or
repairing an old foundation are as follows:
• Vertical post-tensioning to pass through horizontal
crack below;
• A heavy reinforcing grid in the top 18 to 24 in. (450 to
600 mm), capable of carrying the machine load and
serving as a bridge over the old concrete below;
• Horizontal post-tensioning in one or two directions if there
is severe vertical cracking of the remaining concrete;
• Rebuilding of the top foundation with a polymer modified
or similar concrete having tensile and flexural strength
greater than the concrete it replaces;
• Upgrading of the anchor bolts to increase clamping
force, reduce vibration, and reduce the tendency for
cracking near the original bolt termination;
• An adjustable machine support system that allows future
machine realignment to be easily done. This may also be
important if there are uncorrected problems, too costly to
fix, in the foundation mat or subsoil below; and
• Increasing the foundation mass, if required, to five
times the dead weight of the equipment.
Figure 4.15 shows a cross section of a foundation repair
that incorporates several of the above points.
4.7—Sample impedance calculations
This chapter presents numerical calculations using the
equations from the previous sections in Chapter 4. Only the
vertical impedance is included; the other directions follow the
same approach. These calculations are not typical of all
machine foundations and, as such, broad conclusions about
results from alternate equations should not be reached. The
results of these various calculations are tabulated in Table 4.5.
Given quantities:
Soil shear modulus G = 10 k/in.2 = 1,440,000 lbf/ft2;
Soil Poisson’s ratio ν = 0.45;
Soil weight density w = 120 lbf/ft3;
Material damping βm = 5%, or 0.05;
Soil is cohesive;
Mat width a = 20 ft;
Mat length b = 15 ft;
Machine speed = 350 rpm; and
Effective embedment depth = 3 ft.
Base calculations:
Circular operating frequency ωo (definition provided in
Section 3.2.2.1b):
ωo = (350 rpm)(2π rad/rev)(1/60 min/s) = 36.65 rad/s
Equivalent radius for vertical vibration:
R =
ab
------ =
π
(------------------------------20 ft ) ( 15 ft -) = 9.77 ft
π
Eq. (4-9a)
Soil mass density ρ:
ρ = w/gravity = (120 lbf/ft3)/(32.2 ft/s2) = 3.73 lbf-s2/ft4
Because this machine generates a harmonic dynamic force
with a frequency that matches the machine speed, the
frequency of the motion will match the operating speed, ωm
equals ωo.
Nondimensional frequency:
FOUNDATIONS FOR DYNAMIC EQUIPMENT
a o = Rω m ρ ⁄ G
Eq. (4-11)
2
Table 4.5—Summary of vertical impedance
calculations
Calculation
basis
4
( 3.73 lbf-s ⁄ ft )
a o = (9.77 ft)(36.65 rad/s) -------------------------------------------- = 0.576
2
(1,440,000 lbf/ft )
Method 1: Base vertical impedance—Veletsos: Eq. (4-15)
From Table 4.1:
(Values for 0.45 are interpolated between 0.33 and 0.5.)
ν = 0.33
ν = 0.5
ν = 0.45
0.000
0.103
γ1 = 0.350
γ2 = 0.800
0.000
0.235
0.170
0.120
γ3 = 0.000
0.850
0.821
γ4 = 0.750
Parameter χv
No
embedment
kv, k/in. cv, k-s/in.*
Comment
Eq. (4-15c)
adjust by 4.23
8170
7770
110.1
132.4
No material damping
5% material damping
Eq. (4-16c),
Table 4.2
adjust by 4.23
8790
125.3
No material damping
8330
149.3
5% material damping
Eq. (14-3a),
(4-12a)
8530
114.0
No material damping
8530
120.4
5% material damping
ΣWt = 187.5 k
8030
133.5
5% material damping
incl.; “best” calc.
972
37.9
No material damping
833
40.6
5% side material
damping
adjust by
adding 5% for
material
damping
Veletsos and
Verbic (1973)
complete
Eq.
(4-21b)
Embedment
effects (additive to above) adjust by 4.23
*
2
γ1 ( γ2 ao )
χ v = -------------------------- =
2
1 + ( γ2 ao )
351.3R-51
Note: cv values have not been reduced as discussed in Section 4.2.
Eq. (4-15c)
1,321,000 lbf-s/ft, or 110.1 k-s/in.
2
( 0.103 ) [ ( 0.235 ) ( 0.576 ) ]
-------------------------------------------------------------- = 0.00185
2
1 + [ ( 0.235 ) ( 0.576 ) ]
Parameter ψv
2
γ1 γ2 ( γ2 ao )
=
ψ v = --------------------------2
1 + ( γ2 ao )
Eq. (4-15c)
2
( 0.103 ) ( 0.235 ) [ ( 0.235 ) ( 0.576 ) ]
--------------------------------------------------------------------------------- = 0.000436
2
1 + [ ( 0.235 ) ( 0.576 ) ]
Vertical impedance
2
4GR
k v∗ = ---------------- [ ( 1 – χ v – γ 3 a o ) + ia o ( γ 4 + ψ v ) ] Eq. (4-15c)
(1 – ν)
Including material damping through Eq. (4-23) yields
adjusted stiffness and damping terms of
kv(adj) = kv –2β ⋅ cv ⋅ ωm
kv = 98,100 – 2(0.05)(1,321,000)(36.65) = 93,300,000 lb/ft =
7770 k/in.
cv(adj) = cv + 2β × kv/ωm
cv = 1,321,000 + 2(0.05)(98,100,000)/(36.65) = 1,589,000
lbf –s2/ft = 132.4 k – s2/in.
Method 2: Base vertical impedance—Veletsos approximated: Eq. (4-16)/ Table 4.2
Use the same input data as before. Use of Table 4.2 is
allowed because ao is less than 2.0. From Table 4.2, using
cohesive soil, Cv1 = 7.5, Cv2 = 6.8
Vertical impedance
kv* = GR(Cv1 + i aoCv2) = (1,440,000 lbf/ft2) Eq. (4-16c)
(9.77 ft)[7.5 + i (0.576) (6.8)]
2
4 ( 1,440,000 lbf/ft ) ( 9.77 ft )
k v∗ = --------------------------------------------------------------------( 1 – 0.45 )
kv* = 105,500,000 + i 55,100,000 lbf/ft
2
[ ( 1 – 0.00185 – ( 0.12 ) ( 0.576 ) ) + i ( 0.576 ) ( 0.821 + 0.000436 ) ]
k v∗ = 102,318,545 [ 0.9583 + i0.4731 ] = 98,100,000 + i48,400,000 lbf/ft
The stiffness is equivalent to the real part of kv*
kv = 98,100,000 lbf/ft or 8,170 k/in.
Eq. (4-10)
The damping constant is derived from the imaginary part
of kv*
cv = (48,400,000 lbf/ft)/(36.65 rad/s) =
Eq. (4-10)
The stiffness is equivalent to the real part of kv*
kv = 105,500,000 lbf/ft or 8790 k/in.
Eq. (4-10)
The damping constant is derived from the imaginary part
of kv*
cv = 55,100,000/36.65 = 1,503,000 lbf-s/ft
Eq. (4-10)
or 125.3 k-s/in.
Calculations using polynomial expansions of Cv1 and Cv2
support the tabulated constants for this speed of operation
351.3R-52
ACI COMMITTEE REPORT
(ao = 0.576). For higher speeds, the difference between the
constant value and the polynomial are significant.
Including material damping through Eq. (4-23) yields
adjusted stiffness and damping terms of:
kv = 8790 – 2(0.05)(125.3)(36.65) = 8330 k/in.
cv = 125.3 + 2(0.05)(8790)/(36.65) = 149.3 k-s2/in.
Method 3: Base vertical impedance—
Richart-Whitman: Eq. (4-12) and (4-13)
This approach is valid for ao less than 1.0. Assume the
machine weighs 30 kips and the mat is 3.5 ft thick. These
assumptions are not significant for the impedance in the
vertical direction; the kv and cv values are not dependent on
these weights. For rotational motions there is some minor
dependence on the specific assumed values.
The base stiffness is calculated as
kv = 4GRo/(1 – ν) = 4(1,440,000 lbf/ft2)
(4-13a)
(9.77 ft)/(1 – 0.45) = 102,300,000 lbf/ft or 8530 k/in.
c v = 2D v ( k v M ) = 2 ( 0.936 )
2
Method 4: Base vertical impedance—Veletsos and
Verbic (1973)
Veletsos and Verbic (1973) presents more complete
versions of Eq. (4-15) that include material damping. Short
of a complete complex domain solution, this approach is
accepted as the best calculation basis. With material
damping of 5%, those equations yield:
kv = 8030 k/in.
cv = 133.5 k-s/in.
Embedment vertical impedance—Eq. (4-21)
Because the dimensionless frequency (ao = 0.576) is in the
range of 0.5 to 1.5, the use of Table 4.2 factors is permissible.
From Table 4.2, Sv1 = 2.7, Sv2 = 6.7
Embedment vertical impedance
kev* = Gsl [Sv1 + iaoSv2]
The total weight of machine and foundation is
2
( 8530 k/in. ) ( 0.485 k-s /in. ) = 120.4 k-s /in.
(Eq. (4-21b)
W = (20 ft)(15 ft)(3.5 ft)(0.15 kcf) + (30 kips) = 187.5 kips
= (1,440,000 lbf/ft2)(3 ft)[(2.7 + i (0.576)(6.7)]
Either weights (W, w) or masses (m, ρ) can be used in
Eq. (4-12) provided consistency is maintained. The mass
ratio for this calculation is
= 11,660,000 + i16,670,000 lbf/ft
Bv = (1 – ν)W/[4wRo3]
Eq. (4-12a)
= (1 – 0.45)(187.5 kips)/[4(0.12 kcf)(9.77 ft)3]
kv = 11,660,000 lbf/ft, or 972 k/in.
Eq. (4-10)
The damping constant is derived from the imaginary part
of kev*
cv = (16,670,000 lbf/ft)/(36.65 rad/s)
= 0.230
(4-10)
= 455,000 lbf-s/ft, or 37.9 k-s/in.
Damping ratio (geometric damping)
0.425
0.425
D v = ------------- = -------------- = 0.886 = 88.6%
Bv
0.23
The stiffness is equivalent to the real part of kev*
Eq. (4-12a)
The total system mass is
These values can be adjusted for the material damping
effects of the embedment material. The material damping of
this side material may be different from the base material.
Using Eq. (4-23) yields adjusted stiffness and damping terms of
kv = (972 k/in.) – 2 (0.05)(37.9 k-s/in.)(36.65 rad/s)
187.5 kips- = 0.485 k-s 2 /in.
M = W
----- = -------------------------2
g
386.4 in./s
Eq. (4-12a)
= (972 k/in.) – (139.0 k/in.) = 833 k/in.
cv = (37.9 k-s/in.) + 2 (0.05)(972 k/in.)/(36.65 rad/s)
The damping constant is calculated as
= (37.9 k-s/in.) + (2.7 k-s/in.) = 40.6 k-s/in.
c v = 2D v ( k v M ) = 2 ( 0.886 )
2
Eq. (4-14)
2
( 8530 k/in. ) ( 0.485 k-s /in. ) = 114.0 k-s /in.
To include the material damping as a simple addition, the
5% material damping is added to the Dv value, and Eq. (4-14)
is applied. The stiffness is not altered in this approach.
The embedment impedance values are directly additive to
the base impedance values. For example, combining with the
results from the complete solution of Veletsos yields
(kv)total = (kv)base + (kv)embed
= (8030 k/in.) + (833 k/in.) = 8860 k/in.
FOUNDATIONS FOR DYNAMIC EQUIPMENT
(cv)total = (cv)base + (cv)embed
= (133.5 k-s/in.) + (40.6 k-s/in.) = 174.1 k-s/in.
Base impedance values from other calculations could also
be used. Most often, consistent approaches are used. The
overall results of these various calculations are tabulated in
Table 4.5.
Calculation of displacements
Once the stiffness and damping have been determined, the
engineer calculates the maximum response using the approach
in Section 4.3. For demonstration, use the Veletsos and Verbic
values with the embedment terms added and reduce the
damping by 50% as a rough consideration of possible soil
layering effects. Thus, the embedded stiffness is 8860 k/in.,
and the embedded damping is 87 k-s/in. The total system
weight is 187.5 kips. (Refer to Method 3 calculation.)
To calculate motions, determining an excitation force is necessary. Use a given rotor weight of 10 kips and Eq. (3-7) to get
F = 10,000 lbf ⋅ 350 rpm/6000 = 583 lbf
Eq. (3-7)
Applying Eq. (4-44) requires calculating ωn and β.
ωn = (k/m)1/2 = (kg/W) 1/2
ωn = (8860 k/in. ⋅ 32.2 ft/s2 ⋅ 12 in./ft/187.5 k)1/2 = 135.1 rad/s
c
c
β = -------------= --------------2 km
2 kW
----g
87 k-s/in.
β = ------------------------------------------------------------------------------ = 0.663
187.5 k
2 8860 k/in. --------------------------------------------2
32.2 ft/s ⋅ 12 in./ft
F⁄k
A = --------------------------------------------------------------------------------2 2
2
( 1 – ( ω o ⁄ ω n ) ) + ( 2βω o ⁄ ω n )
Eq. (4-44)
583 ⁄ 8860
A = -------------------------------------------------------------------------------------------------------------------------------------2 2
2
( 1 – ( 36.65 ⁄ 135.1 ) ) + ( 2 ( 0.663 ) ( 36.65 ) ⁄ ( 135.1 ) )
0.0658 mils
= ---------------------------0.994
The predicted amplitude motion of this SDOF system is
0.0662 mils or 0.0000662 in. (0.00168 mm). The peak-topeak motion is 0.1324 mils (3.36 mm) at 350 rpm. For
comparison using Fig. 3.10, this motion at this speed is seen
to qualify as “extremely smooth.” Similarly on Fig. 3.11, the
resultant motion at the operating speed is well below the
various corporate and other standards for acceptance.
351.3R-53
Although a ωn value is computed for use in Eq. (4-44), this
is not actually the system natural frequency because the value
has been computed using a stiffness value specific to motion at
a specific frequency (350 rpm or 36.65 rad/s). With frequency
dependent impedance, as the frequency of excitation increases,
the stiffness decreases. Determining the frequency at which
maximum response occurs requires an iterative solution across
a range of frequencies. For this problem, such an iteration
shows that the maximum undamped response (comparable
with the aforementioned ωn value) occurs at a speed of
1030 rpm (108 rad/s) and the more realistic, maximum
damped response occurs at 1390 rpm (146 rad/s).
CHAPTER 5—CONSTRUCTION CONSIDERATIONS
5.1—Subsurface preparation and improvement
5.1.1 General considerations—Equipment foundations
can be supported on soil, rock, piles, or drilled piers,
depending on the geotechnical conditions of the site. The
engineer and geotechnical consultant determine the extent of
soil investigation and subsurface preparation, which may
vary from minimal to extensive. The construction contractor
executes the subsurface preparation and the geotechnical
engineer verifies it. Adjustments are generally made as
required as work progresses.
The engineer should consider the effect of vibratory and
impulsive loading on the underlying soils to determine if
they are susceptible to dynamic consolidation, particularly
under foundations of large dynamic equipment. Thus, additional soil parameters, such as shear wave velocity, dynamic
modulus of elasticity, and Poisson’s ratio, may be required
from the soil investigation to perform a dynamic analysis
(Chapter 3).
5.1.2 Specific subsurface preparation and improvements—The contractor should prepare the site in a way
consistent with the assumptions made and parameters used
in the foundation analysis. Due to the dynamic nature of
loads acting on the foundations, the contractor should pay
particular attention to proper compaction and consolidation
of the soils.
Specific subsurface preparation and related treatment may
be required for one or more of the following reasons:
• If the exploratory borings, field tests and observations,
and subsequent laboratory tests dictate the necessity of
a subsurface treatment;
• If the exploratory borings reveal nonuniform and heterogeneous conditions with irregularities requiring local
remedies; and
• If close inspection of the foundation excavation indicates
conditions other than the ones extrapolated from the
borings, thereby requiring special preparation and
treatment—generally of a localized nature.
Common site-specific subsurface preparations and treatment for these conditions are:
a) Unstable excavation slopes—Unstable slopes may be
stabilized by flattening the slope, benching, dewatering,
shoring, freezing, injection with chemical grouts, or
supporting with dense slurries;
351.3R-54
ACI COMMITTEE REPORT
b) Stratification—Excavations with slopes parallel to the
direction of stratification are avoided by flattening the slope
or by providing adequate shoring;
c) Wet excavation—During construction, groundwater is
normally lowered below the bottom level of the excavation;
deep well pumps or well points are commonly used. Another
method is to create an impervious barrier around the excavation
with cofferdams or caissons, chemical grout injection, sheet
piles, or slurry trenches. A sump pit collects groundwater
intrusion. The selection of an appropriate method depends
on the characteristics of the surface soils encountered, costs,
and the preferences of the constructor;
d) Small surface pockets of loose sand—Loose sand
pockets are normally compacted to the degree of specified
compaction. Alternatively, if the predominant soil is hard,
the loose sand may be removed and replaced with flowable
fill. Loose sand under dynamically loaded foundations is
particularly prone to differential settlement and should be
eliminated during construction;
e) Large deposits of loose sands—The loose sands may be
stabilized by vibrofloatation or dynamic consolidation,
whichever offers an economic advantage. A prediction of
long-term settlements considering the vibratory loads may
be necessary;
f) Presence of organic material or unconsolidated soft
clays—All organic materials and soft clays are normally
removed and replaced with suitable, well-compacted fill that
provides the characteristics desired for the proper performance of the foundation. Alternatively, piling or drilled piers
may be used to carry foundation loads to sound bearing strata;
g) Fissured rock—The extent of fissures is evaluated to
determine if remedial treatment is needed. Pressure grouting
is a suitable remedy for some types of fissures. In the case of
seismic faults, thorough geotechnical and geological evaluation is required to ascertain the potential hazard. Where
significant hazards exist, relocation of the entire facility to
avoid the hazard is a suitable remedy;
h) Irregularly weathered rock—The weathered seams are
cleaned and replaced with lean concrete. Alternatively, the
foundation may be revised to reach sound rock;
i) Solution cavities in limestone deposits—The voids, if
small, are pumped full of grout or, in the case of large holes,
lean concrete under a pressure head;
j) Unconsolidated clay—Clays may be preloaded and
related settlements monitored. (Early identification is important to gain lead time and avoid slippage in the construction
schedule.) Alternatively, piling or drilled piers may be used
to carry foundation loads to firm bearing strata; and
k) Cold climates—The construction crew should not place
foundations on fine-grained soils subject to frost heave. The
crew should provide proper drainage by placing a freedraining sand or gravel layer under the foundation to mitigate the possibility of frost heave where such hazard exists.
As an alternative, the bottom of the foundation is placed
below the frost line.
5.2—Foundation placement tolerances
Foundation placement tolerances depend largely on the
type of equipment being supported and are specified by the
engineer on the drawings or in the specifications. The
construction crew should use templates during concrete
placement to support anchor bolts and other embedments
that must be precisely positioned.
5.3—Forms and shores
5.3.1 General requirements for forms—Forms and shoring
for construction of concrete foundations should follow the
recommendations of ACI 347R. As applicable, provisions of
ACI 301 should be specified.
5.3.2 Shoring—Shoring should support the concrete loads,
impact loads, and temporary construction loads. Transverse and
longitudinal bracing may be required to sustain lateral forces.
The formwork engineer should consider wind loads in the
shoring design. It is not usually necessary to consider
seismic loads due to the limited time shoring is in place. A
licensed professional engineer should prepare the design of
the formwork and submit it to the design engineer for review.
5.3.3 Shoring systems and formwork for large elevated
foundations—For large equipment foundation pedestals,
such as turbine-generator foundations, temporary formwork
systems are generally used. Less frequently, permanent
systems may be used for special applications. The contractor
usually selects a temporary support system. The selection is
influenced by the erection sequence of the building (if the
equipment is enclosed), the equipment installation procedure, and access requirements at the time of placement of the
foundation. Some of the permanent systems may affect the
design and cost of the foundation. Therefore, the design
engineer may wish to consult with building contractors
before deciding on a permanent formwork system.
Some of the systems used are:
• Standard construction shoring consisting of temporary
shore legs supported by the foundation mat and supporting
the soffit forms of a foundation deck;
• Shoring consisting of structural steel beams supported
on brackets attached to the foundation columns. The
forms rest on top of the beams. Jacking devices are
used to lower the beams and forms for removal after the
concrete reaches sufficient strength;
• Embedded structural steel shapes (rolled wide flange
beams, girders, angles, or channels) supported on the
foundation columns and carrying the permanent deck
forms. The forms (steel decking) usually rest on the
bottom flanges of the steel shapes. Because the steel
shapes are embedded in the foundation deck, the design
engineer should to be careful to avoid interferences
with the reinforcing bars and with other embedments
(anchor bolts, plates, pipe sleeves and conduits);
• Embedded structural steel trusses supported on the
foundation columns and carrying the permanent deck
forms on the bottom chords. The trusses, if specially
designed, can also be considered as reinforcing to carry
the operating loads acting on the foundation deck.
Checking for interferences between the trusses and the
FOUNDATIONS FOR DYNAMIC EQUIPMENT
reinforcing bars and other embedments is important to
avoid serious construction problems; and
• Precast concrete deck forms supported by the foundation
columns. These can be flat bottom “U” and double “U”
shapes.
All of these systems, except the Standard Construction
Shoring System, allow early access under the foundation
deck. The standard shoring system, however, has the least
impact on the foundation design. The remaining four
systems should be coordinated, in varying degrees, with the
foundation design.
In all five cases, the design engineer should review the
contractor’s construction procedure.
5.4—Sequence of construction and
construction joints
Many large machine foundations are too massive for the
concrete to be placed in one continuous operation. Construction
joints subdivide large foundations into smaller placement
units. Subdivision of large foundations by construction joints
also helps reduce internal heat of hydration in concrete and
shrinkage cracks in the foundation. To gain maximum
benefit, the constructor should place alternate foundation
segments and allow them cure and shrink as long as the
construction schedule permits before the intervening
segments are placed.
The structural integrity of the foundation requires that
joints be carefully constructed using accepted practices for
construction joints in major concrete structures, such as
specified in ACI 301. Project specifications normally require
that the constructor obtain the approval of the engineer for
construction joint locations and details.
The location of construction joints should follow normal
reinforced concrete building practice. Joints in columns should
be located at or near the floor line and at the underside for
supported beams. If the beams are haunched, the joints should
be located at the underside of the deepest haunch. Joints in
beams and mats should be located at sections of low stress.
Transverse construction joints should be at right angles to
the main reinforcement. Horizontal joints in beams and slabs
placed in more than one lift should be supplied with sufficient
transverse reinforcing to develop the required horizontal shear
capacity by shear friction. Preparation of construction joints
should be in accordance with ACI 304R.
Transfer of loads across a construction joint should be
provided for by specific means. Tensile loads, for example,
should be transferred by extending reinforcing bars across the
joint. Transfer of compressive loads can be accomplished by
ensuring that the concrete on both sides of the joint is strong
and dense. Additional measures are needed to transfer shear
loads. Shear keys should be cast in the face of the joint. Alternatively, the face of the joint can be roughened sufficiently for
shear loads to be transferred by shear friction. With the latter
method, sufficient reinforcing bars should extend across the
joint to hold the surfaces of the joint in close contact.
351.3R-55
5.5—Equipment installation and setting
5.5.1—Shims, wedges, and bolts represent a typical interface
system between the foundation and the machine base. The
chosen interface system can be influenced by the machine
manufacturer’s recommendations and requirements, the
foundation construction procedures, the setting and adjustment
of the equipment, and the final tolerances.
Shims, which are usually carbon steel or brass stock in various
thicknesses, have both economical and high load-bearing
qualities. Shims should be fabricated with rounded corners.
Wedges are usually the double-wedge type and are offered
by several mounting-equipment manufacturers. The double
wedge mount often has one or more threaded studs for
precise vertical adjustment and for locking the sliding wedge
into the required position. A lock nut may also be used for
locking the main horizontal stud into the final position.
Other types of wedges often used by millwrights include
various shaped temporary steel wedges. Temporary wedges
are usually tolerance adjustment tools placed before
grouting, and they are removed after the hardening of the
grout material. Permanent wedge assemblies allow future
adjustments on ungrouted equipment bases.
The manufacturer’s drawings should give the required bolt
diameters. For construction, the design drawings or specifications should provide bolt diameters, types, overall lengths,
threaded lengths, projections, materials, the method of bolt
tightening, and required torques. When a specific preload is
required by the manufacturer or design engineer, Eq. (5-1)
can be used to determine the bolt torque
Tb = WpKndn
(5-1)
where
Tb = bolt torque, lbf-in. (N-m);
= nominal bolt diameter, in.;
dn
Wp = preload, lbf (Section 4.5.2); and
Kn = nut factor for bolt torque (dimensionless).
Typical values of Kn (often in the range of 0.1 to 0.3) are
tabulated in Bickford (1995) and vary with the lubrication
and condition of the bolts. Special coatings may require
manufacturer’s data.
Required bolt tightening can be accomplished with a posttensioning jacking procedure, a turn-of-the-nut method, or a
calibrated wrench. Post-tensioning jacking is used on the
deeper anchorages with nonbonded shanks. When the shank
length is embedded in concrete, the turn-of-the-nut method
or sequential calibrated wrench tightening is specified.
Section 4.4.2.4 presents comments on monitoring the bolt
tension. Impact wrenches are not used for tightening a bolt
component when part of the anchorage is embedded in
concrete because of the extremely high torque and tensile
forces delivered by such tools.
5.5.2—Embedments in the concrete include the anchor
bolt assemblies previously described, shear lugs, and shear
transferring devices. Because shear is a component of a total
load transferred to the concrete foundation, steel lugs can be
integral parts of the machine base. Such lugs are grouted into
shear key grooves previously cast into the concrete base.
351.3R-56
ACI COMMITTEE REPORT
5.6—Grouting
5.6.1 Types of grout—There are two basic types of grout:
cementitious (cement-based) grouts and polymer (including
epoxy-based) grouts. Cementitious grouts are lower in cost,
but polymer grouts have higher resistance to chemicals,
shock, and vibratory loads.
5.6.2 Applications—ACI 351.1R contains details on the
application of grouts. In specifying grout systems, the engineer should consider the different characteristics of each
type of grout along with field limitations and match these
with specific requirements of the job. In evaluating which
cementitious grout should be used, the engineer should
consider the placability of the grout and its physical properties:
volume change, compressive strength, working time, consistency, and setting time. In evaluating polymer grouts, the
engineer should consider the same factors along with creep
and the effects of temperature-induced volume changes. The
engineer should review the design of the equipment base,
accessibility of the grouting location, clearances provided
for the grout, and design of anchor bolts. Most of the grouts
on the market are premixed, prepackaged materials and contain
manufacturer’s instructions on surface preparation, formwork, mixing, placing, and curing procedures.
5.7—Concrete materials (ACI 211.1, ACI 301)
Large machine foundations require special attention to the
design and control of the concrete mixture (ACI 207.1R and
ACI 207.4R).
Many foundations are massive enough for the heat of
hydration of the cement to generate a large thermal differential
between the inside and the outside, which may cause
unacceptable surface cracking unless steps are taken to
reduce the rate of release of this heat. Creep, differential
thermal expansion, and shrinkage can cause distortion of the
foundation and result in unacceptable changes in machine
alignment. It is important to design the concrete mixture to
minimize creep and shrinkage and to reduce the thermal
expansion of the hardened concrete. Temperatures may be
monitored using thermocouples or resistance thermometers.
If excessive temperatures are detected, surface cooling
systems can be used to provide limited benefits in controlling
temperatures (ACI 207.4R). Expansive reaction of the
concrete aggregate with alkalies in the cement can be
avoided by proper choice of cement and aggregate.
To minimize the rate of release of the heat of hydration,
and to control shrinkage and creep, the following steps are
normally followed:
• The lowest content of cementitious material consistent
with attaining the required strength and durability is used;
• Part of the cement is replaced with fly ash or other
pozzolan;
• The placing temperature of fresh concrete is lowered by
chilling the aggregate, using chipped ice for mixing
water, or both;
• The largest practical size aggregate is used to allow further
reduction in the amount of cement;
• Moderate heat cement (Type II) is used;
• A water-reducing agent is used to allow further reduction
of the cement factor;
• Low slump and effective vibration are used;
• Concrete placement by pumps, which require concrete
mixtures having high amounts of cement and small
aggregate sizes, is avoided; and
• Sizes of placements for large foundations are reduced.
High-range water-reducing admixtures may be an appropriate choice because they are consistent with their general
applicability to mass concrete applications and heavily
reinforced installations for which workability is an issue.
The coefficient of thermal expansion of the hardened
concrete can be controlled by the choice because they are of
aggregates and because it primarily depends on the coefficient
of thermal expansion of the aggregate. When excessive thermal
expansion may be a problem, the coefficient of expansion of
available aggregates is measured to determine their suitability
for the application. (In many regions there may be very limited
choices in the types and sources of aggregates.)
Expansion of concrete from alkali-aggregate reaction can
be minimized by using a low-alkali cement, by replacing a
portion of the cement with a fly ash or nonfly ash pozzolan
meeting the requirements of ASTM C 618, and by selecting
low-reactivity aggregates. The potential reactivity of
aggregates can be evaluated with the procedures and tests
described in ASTM C 295, ASTM C 227, ASTM C 289, and
ASTM C 586. ASTM C 33 and ACI 225R cover the evaluation
methods of the potential reactivity of aggregates.
The cement content should be low enough to help meet
heat of hydration requirements but high enough to meet
strength, creep, and shrinkage requirements. (It may not be
possible to completely solve the heat problem by reducing
the heat of hydration. Cooling, small placements, or
pozzolans may also be needed.)
5.8—Quality control
Because the foundation for the machine acts as an integral
part of the machine-foundation-soil system, an appropriate
quality control program should be implemented to ensure
that the design requirements are met during construction.
ACI 311.4R contains guidance on items to include in the
quality program. ACI 311.5R contains guidance on concrete
plant inspection and testing of ready-mixed concrete. ACI SP-2
contains general guidance on inspection of concrete. The
quality control program should include requirements for
control of material quality, the engineer’s approval of critical
construction procedures, and onsite verification of compliance with design drawings and project specifications by a
qualified field engineer, preferably certified by ACI as a
concrete construction inspector.
Design drawings and project specifications should provide
foundation requirements to the constructor and field engineer.
The field engineer typically is required to report to the design
engineer or owner any changes or modifications to the specified design warranted by the conditions in the field. The
design engineer or owner should approve any changes in the
specified design and document them in accordance with
preestablished procedures.
FOUNDATIONS FOR DYNAMIC EQUIPMENT
The quality control program and inspections should be thoroughly documented and be available for the owner’s reviews.
The quality control program should be consistent with those
commonly implemented for construction projects of similar
importance. ACI 301 can be cited as part of that quality control
program. Laboratories providing testing and inspections should
be accredited to the requirements of ASTM E 329.
CHAPTER 6—REFERENCES
6.1—Referenced standards and reports
Documents and standards produced by national and international organizations that are relevant and referenced in this
report are listed as follows. In the preparation of this report,
currently available editions of these documents were used.
Because these documents are subject to frequent revision,
the reader is advised to contact the sponsoring agency for the
latest versions.
Acoustical Society of America (ASA)
ASA/
Mechanical Vibration—Balance Quality
ANSI S2.19
Requirements of Rigid Rotors, Part 1:
Determination of Permissible Residual
Unbalance
American Concrete Institute (ACI)
ACI 117
Standard Specification for Tolerances for
Concrete Construction and Materials
ACI 116R
Cement and Concrete Terminology
ACI 121R
Quality Management System for Concrete
Construction
ACI 207.1R
Mass Concrete
ACI 207.2R
Cracking of Massive Concrete
ACI 207.4R
Cooling and Insulating Systems for Mass
Concrete
ACI 211.1
Standard Practice for Selecting Proportions
for Normal, Heavyweight, and Mass
Concrete
ACI 215R
Considerations for Design of Concrete
Structures Subjected to Fatigue Loading
ACI 225R
Guide to the Selection and Use of Hydraulic
Cements
ACI 301
Specifications for Structural Concrete
ACI 304R
Guide for Measuring, Mixing, Transporting, and Placing Concrete
ACI 307/307R Design and Construction of Reinforced
Concrete Chimneys and Commentary
ACI 311.4R
Guide for Concrete Inspection
ACI 311.5R
Guide for Concrete Plant Inspection and
Field Testing of Ready-Mixed Concrete
ACI 318/318R Building Code Requirements for Structural
Concrete and Commentary
ACI 347R
Guide to Formwork for Concrete
ACI 349.1R
Reinforced Concrete Design for Thermal
Effects on Nuclear Power Plant Structures
ACI 351.1R
Grouting Between Foundations and Bases
for Support of Equipment and Machinery
ACI 351.2R
Foundations for Static Equipment
ACI SP-2
ACI Manual of Concrete Inspection
351.3R-57
American Petroleum Institute (API)
API 541
Form-Wound Squirrel Cage Induction
Motors—250 Horsepower & Larger
API 610
Centrifugal Pumps for Petroleum, Heavy
Duty Chemical, & Gas Industry Services
API 612
Special-Purpose Steam Turbines for Petroleum, Chemical, & Gas Industry Services
API 613
Special-Purpose Gear Units for Petroleum,
Chemical, & Gas Industry Services
API 617
Centrifugal Compressors for Petroleum,
Chemical, and Gas Industry Services
API 618
Reciprocating Compressors for Petroleum,
Chemical, and Gas Industry Services
API 619
Rotary-Type
Positive
Displacement
Compressors for General Refinery Services
API 684
Tutorial on the API Standard Paragraphs
Covering Rotor Dynamics and Balance (An
Introduction to Lateral Critical and Train
Torsional Analysis and Rotor Balancing)
API 686
Recommended Practice for Machinery
Installation and Installation Design
American Society of Civil Engineers (ASCE)
ASCE 7
Minimum Design Loads for Buildings and
Other Structures
ASTM International
ASTM A 36/ Standard Specification for Carbon Structural
A 36M
Steel
ASTM A 193 Standard Specification for Alloy-Steel and
Stainless Steel Bolting Materials for HighTemperature Service
ASTM A 307 Standard Specification for Carbon Steel
Bolts and Studs, 60,000 PSI Tensile
Strength
ASTM A 615 Standard Specification for Deformed and
Plain Billet-Steel Bars for Concrete Reinforcement
ASTM C 33
Standard Specification for Concrete
Aggregates
ASTM C 227 Standard Test Method for Potential Alkali
Reactivity of Cement-Aggregate Combinations (Mortar-Bar Method)
ASTM C 289 Standard Test Method for Potential AlkaliSilica Reactivity of Aggregates (Chemical
Method)
ASTM C 295 Standard Guide for Petrographic Examination
of Aggregates for Concrete
ASTM C 580 Standard Test Method for Flexural Strength
and Modulus of Elasticity of ChemicalResistant Mortars, Grouts, Monolithic
Surfacings, and Polymer Concretes
ASTM C 586 Standard Test Method for Potential Alkali
Reactivity of Carbonate Rocks for Concrete
Aggregates (Rock Cylinder Method)
ASTM C 618 Standard Specification for Coal Fly Ash
and Raw or Calcined Natural Pozzolan for
Use as a Mineral Admixture in Concrete
351.3R-58
ACI COMMITTEE REPORT
ASTM C 1181 Standard Test Methods for Compressive
Creep of Chemical-Resistant Polymer
Machinery Grouts
ASTM D 4015 Standard Test Methods for Modulus and
Damping of Soils by the Resonant Column
Method
ASTM E 329 Standard Specification for Agencies
Engaged in the Testing and/or Inspection of
Materials Used in Construction
ASTM F 1554 Standard Specification for Anchor Bolts,
Steel, 36, 55, and 105-ksi Yield Strength
ASTM SI10
American National Standard for Use of the
International System of Units (SI): The
Modern Metric System
Deutsches Institut für Normung (DIN)
DIN 4024 Part 1 Machine
Foundations:
Elastic
Supporting Constructions for Machines
with Rotating Masses
DIN 4024 Part 2 Machine Foundations: Rigid Supporting
Constructions for Machines with Periodic
Excitation
DIN 4025
Foundations for Drop Forging Machinery
DIN 4150 Part 1 Vibrations in Buildings: Prediction of
Vibration Parameters
DIN 4150 Part 2 Vibrations in Buildings: Effects on
Persons in Buildings
DIN 4150 Part 3 Vibrations in Buildings: Effects on
Structures
Federal Emergency Management Administration (FEMA)
FEMA 302
NEHRP Recommended Provisions for
Seismic Regulations for New Buildings and
Other Structures, 1997
International Conference of Building Officials (ICBO)
IBC
International Building Code
UBC
Uniform Building Code
International Standards Organization (ISO)
ISO 1940-1
Mechanical Vibration—Balance Quality
Requirements of Rigid Rotors—Part 1:
Determination of Permissible Residual
Unbalance
ISO 2631-1
Mechanical Vibration and Shock—Evaluation of Human Exposure to Whole Body
Vibration—Part 1: General Requirements
ISO 10816-1
Mechanical
Vibration—Evaluation
of
Machine Vibration by Measurements on NonRotating Parts—Part 1: General Guidelines
ISO 10816-2 Mechanical Vibration—Evaluation of
Machine Vibration by Measurements on
Non-Rotating Parts—Part 2: Large LandBased Steam Turbine Generator Sets in
Excess of 50 MW
ISO 10816-3
Mechanical
Vibration—Evaluation
of
Machine Vibration by Measurements on
Non-Rotating Parts—Part 3: Industrial
ISO 10816-4
ISO 10816-5
ISO 10816-6
Machines with Nominal Power Above 15 kW
and Nominal Speeds Between 120 r/min and
15,000 r/min when Measured In Situ
Mechanical Vibration—Evaluation of
Machine Vibration by Measurements on
Non-Rotating Parts—Part 4: Gas Turbine
Driven Sets Excluding Aircraft Derivatives
Mechanical Vibration—Evaluation of
Machine Vibration by Measurements on
Non-Rotating Parts—Part 5: Machine Sets
in Hydraulic Power Generating and
Pumping Plants
Mechanical
Vibration—Evaluation
of
Machine Vibration by Measurements on NonRotating Parts—Part 6: Reciprocating
Machines with Power Ratings above 100 kW
Verein Deutscher Inginieure (VDI)
VDI 2057
Effect of Mechanical Vibrations on Human
Beings
These publications may be obtained from these organizations:
Acoustical Society of America
335 East 45th St.
New York, NY 10017-3483
web: http://asa.aip.org
American Concrete Institute
38800 Country Club Dr.
Farmington Hills, MI 48331
Web: http://www.concrete.org
American Petroleum Institute
Order Desk
1220 L St. NW
Washington, DC 20005-4070
Web: http://www.api.org
American Society of Civil Engineers
1801 Alexander Bell Dr.
Reston, VA 20191
Web: http://www.asce.org
ASTM International
100 Barr Harbor Dr.
West Conshohocken, PA 19428
Web: http://www.astm.org
Deutsches Institut für Normung (DIN)
Burggrafenstrasse 6
DE-10772 Berlin
Germany
Web: http://www.din.de
Federal Emergency Management Agency
Building Seismic Safety Council
1090 Vermont Avenue, Suite 700
Washington, D.C. 20005
Web: http://www.bssconline.org
FOUNDATIONS FOR DYNAMIC EQUIPMENT
International Conference of Building Officials
5360 Workman Mill Road
Whittier, CA 90601-2298
Web: http://www.icbo.org
International Standards Organization
1, rue de Varembé
Case postale 56
CH-1211 Genéve 20
Switzerland
Web: http://www.iso.ch
Verein Deutscher Inginieure (VDI)
P.O. Box 10 11 39
D-40002 Dusseldorf
Germany
Web: http://www.vdi.de
6.2—Cited references
Arya, S. C.; O’Neill, M. W.; and Pincus, G., 1979, Design
of Structures and Foundations for Vibrating Machines, Gulf
Publishing Co., Houston, Tex.
Baxter, R. L., and Bernhard, D. L., 1967, “Vibration Tolerances for Industry,” ASME Paper 67-PEM-14, Plant Engineering and Maintenance Conference, Detroit, Mich., Apr.
Beredugo, Y. O., and Novak, M., 1972, “Coupled Horizontal
and Rocking Vibration of Embedded Footings,” Canadian
Geotechnical Journal, NRCC, V. 9, No. 4, pp. 477-497.
Bickford, J. H., 1995, An Introduction to the Design and
Behavior of Bolted Joints, New York and Basel, Marcel
Dekker, Inc., 3rd Edition, 992 pp.
Blake, M. P., 1964, “New Vibration Standards for Maintenance,” Hydrocarbon Processing and Petroleum Refiner, Gulf
Publishing Co, Houston, Tex., V. 43, No. 1, Jan., pp. 111-114.
Bowles, J. E., 1996, Foundation Analysis and Design, 5th
Edition, McGraw-Hill, New York, 1024 pp.
Das, B. M., 1993, Principles of Soil Dynamics, Boston,
Mass., PWS-Kent Publishers, 592 pp.
Elsabee, F., and Morray, J. P., 1977, “Dynamic Behavior
of Embedded Foundations,” Research Report R77-33, Civil
Engineering Department, Massachusetts Institute of Technology, Cambridge, Mass.
EPRI, 1980 “Dynamics of Power Plant Fan-Foundation
Systems,” Final Report CS-1440, July, p. 3-12.
Fang, H.-Y., 1991, Foundation Engineering Handbook,
2nd Edition, Van Nostrand Reinhold, New York, 924 pp.
Gazetas, G., 1983, “Analysis of Machine Foundation
Vibrations: State of the Art,” Soil Dynamics and Earthquake
Engineering, V. 2, No. 1, Jan., pp. 2-42.
GERB, 1995, Vibration Isolation Systems, 9th Edition,
GERB Vibration Control Systems, Inc., Lisle, Ill.
Hardin, B. O., and Black, W. L., 1968, “Vibration
Modulus of Normally Consolidated Clay,” Journal of Soil
Mechanics and Foundations, ASCE, V. 92, No. SM2, Mar.,
pp. 353-369.
Hardin, B. O., and Richart, F. E., Jr., 1963, “Elastic Wave
Velocities in Granular Soils,” Journal of Soil Mechanics and
Foundations, ASCE, V. 89, No. SM1, Feb., pp. 33-65.
351.3R-59
Harris, C. M., 1996, Shock and Vibration Handbook, 4th
Edition, McGraw-Hill, New York, 1456 pp.
Karabalis, D. L., and Beskos, D. E., 1985, “Dynamic
Response of 3-D Embedded Foundations by Boundary
Element Method,” 2nd Joint ASCE/ASME Mechanics
Conference, Albuquerque, N.M., 34 pp.
Kausel, E., and Ushijima, R., 1979, “Vertical and
Torsional Stiffness of Cylindrical Footings,” Research
Report R79-6, Civil Engineering Department, Massachusetts Institute of Technology, Cambridge, Mass.
Kobayashi, S., and Nishimura, N., 1983, “Analysis of
Dynamic Soil-Structure Interactions by Boundary Integral
Equation Method,” 3rd International Symposium on Numerical Methods in Engineering, Paris, pp. 342-353.
Kuhlmeyer, R. L., 1979a, “Vertical Vibration of Piles,”
Journal of Geotechnical Engineering, ASCE, V. 105, No. GT2,
Feb., pp. 273-287.
Kuhlmeyer, R. L., 1979b, “Static and Dynamic Laterally
Loaded Floating Piles,” Journal of Geotechnical Engineering, ASCE, V. 105, No. GT2, Feb., pp. 289-304.
Lakshmanan, N., and Minai, R., 1981, “Dynamic Soil
Reactions in Radially Non-homogeneous Soil Media,”
Bulletin of the Disaster Prevention Research Institute, Kyoto
University, Part 2, V. 31, No. 279, pp. 79-114.
Lifshits, A.; Simmons, H. R.; and Smalley, A. J., 1986,
“More Comprehensive Vibration Limits for Rotating
Machinery,” Journal of Engineering for Gas Turbines and
Power, ASME , V. 108, No. 4, Oct., pp. 583-590.
Mandke, J. S., and Smalley, A. J., 1989, “Foundation
Thermoelastic Distortion,” Report No. 89-3, Pipeline and
Compressor Research Council.
Mandke, J. S., and Smalley, A. J., 1992, “Thermal Distortion
of Reciprocating Compressor Foundation Blocks,” ASME
Paper No. 92-Pet-3, ASME Energy-Sources Technology
Conference and Exhibition, Jan. 26-30, Houston, Tex.
Mandke, J. S., and Troxler, P. J., 1992, “Dynamics of
Compressor Skids,” GMRC Technical Report No. TR 92-2,
Gas Machinery Research Council, Mar.
Mitwally, H., and Novak, M., 1987, “Response of Offshore
Towers with Pile Interaction,” Journal of Engineering
Mechanics, ASCE, V. 113, No. 7, July, pp. 1065-1084
Novak, M., 1970, “Prediction of Footing Vibrations,”
Journal of the Soil Mechanics and Foundations Division,
ASCE, V. 96, No. SM3, May, pp. 837-861.
Novak, M., 1974, “Dynamic Stiffness and Damping of
Piles,” Canadian Geotechnical Journal, NRCC, V. 11, No. 4,
pp. 574-598.
Novak, M., 1977, Soil-Pile-Foundation Interaction,
Proceedings of the 9th International Conference on Soil
Mechanics and Foundation Engineering, Japanese Society
of Soil Mechanics and Found Engineering, Tokyo, Japan,
July 11-15., pp. 309-315.
Novak, M., 1979, “Soil Pile Interaction Under Dynamic
Loads,” Numerical Methods in Offshore Piling: Proceedings
of a Conference Organized by the Institution of Civil Engineers
and Held in London, May 22-23, ICE, London, pp. 59-68.
Novak, M., and Beredugo, Y.O., 1972, “Vertical Vibration of Embedded Footings,” Journal of the Soil Mechanics
351.3R-60
ACI COMMITTEE REPORT
and Foundations Division, ASCE, V. 98, No. SM12, Dec.,
pp. 1291-1310.
Novak, M., and Sachs, K., 1973, “Torsional and Coupled
Vibrations of Embedded Footings,” International Journal of
Earthquake Engineering and Structural Dynamics, V. 2, No. 1,
pp. 11-33.
Novak, M., and Sheta, M., 1980, “Approximate Approach
to Contact Problems of Piles,” Dynamic Response of Pile
Foundations, Analytical Aspects: Proceedings of a Session
Sponsored by the Geotechnical Engineering Division at the
ASCE National Convention, Oct. 30, M. W. O’Neill and R.
Dobry, eds., ASCE, New York.
Novak, M., and Sheta, M., 1982, “Dynamic Response of
Piles and Pile Groups,” Proceedings of the 2nd International
Conference on Numerical Methods in Offshore Piling, Apr.
29-30, University of Texas, Austin, Tex.
Pantermuehl, P. J., and Smalley, A. J., 1997a,
“Compressor Anchor Bolt Design,” GMRC Technical
Report No. TR 97-6, Gas Machinery Research Council, Dec.
Pantermuehl, P. J., and Smalley, A. J., 1997b, “Friction
Tests—Typical Chock Materials and Cast Iron,” GMRC
Technical Report No. TR 97-3, Gas Machinery Research
Council, Dec.
Poulos, H. G., 1979, “Group Factors for Pile-Deflection
Estimation,” Journal of the Geotechnical Engineering
Division, ASCE, V. 105, No. 12, pp. 1489-1509.
Poulos, H. G., and Davis, E. H., 1980, Pile Foundation
Analysis and Design, John Wiley and Sons, New York., 410 pp.
Poulos, H. G., and Randolph, M. F., 1983, “Pile Group
Analysis: A Study of Two Methods,” Journal of Geotechnical Engineering, ASCE, V. 109, No. 3, Mar., pp. 355-372.
Randolph, M. F., 1981, “The Response of Flexible Piles
to Lateral Loading,” Geotechnique, ICE, V. 31, No. 2,
June, pp. 247-259.
Randolph, M. F., and Poulos, H. G., 1982, “Estimating the
Flexibility of Offshore Pipe Groups,” Proceedings of the 2nd
International Conference on Numerical Methods in Offshore
Piling, Apr. 29-30, 1982, University of Texas, Austin, Tex.
Richart, F. E., Jr.; Hall, J. R., Jr.; and Woods, R. D., 1970,
Vibrations of Soils and Foundations, Prentice-Hall, Englewood Cliffs, N.J., 414 pp.
Richart, F. E., Jr., and Whitman, R. V., 1967, “Comparison of Footing Vibration Tests with Theory,” Journal of
Soil Mechanics and Foundations Division, ASCE, V. 93,
No. SM6, Nov., pp. 143-168.
Schenck, T., 1990, Fundamentals of Balancing, Schenck
Trebel Corporation, Deer Park, N.Y.
Seed, H. B., and Idriss, I. M., 1970, “Soil Moduli and
Damping Factors for Dynamic Response Analysis,” Report
70-1, EERC, Dec., Berkeley, Calif.
Sharnouby, E., and Novak, M., 1985, “Static and Low
Frequency Response of Pile Groups,” Canadian Geotechnical Journal, NRCC, V. 22, No. 1, Jan., pp. 79-94.
Smalley, A. J., 1985, “Topical Report: Misalignment and
Temperature Measurements on a Fully Grouted Reciprocating Compressor,” Project PR15-174, American Gas
Association.
Smalley, A. J., 1988, “Dynamic Forces Transmitted by a
Compressor to its Foundation,” ASME Conference Paper
(preprint), New Orleans, La., Jan. 10-14.
Smalley, A. J., 1997, “Epoxy Chock Material Creep
Tests,” GMRC Technical Report No. TR 97-5, Gas
Machinery Research Council, Dec. 1997.
Smalley, A. J. and Harrell, J. P., 1997, “Foundation
Design,” presented at the 1997 PCRC Gas Machinery Conference, October 7, 1997, Austin, Tex., Pipeline and Compressor
Research Council. The Pipeline and Compressor Research
Council (PCRC) is now the Gas Machinery Research Council,
(GMRC). Website: http://www.gmrc.org
Smalley, A. J., and Pantermuehl, P. J., 1997, “Foundation
Guidelines,” GMRC Technical Report No. TR 97-2, Gas
Machinery Research Council, Jan. 1997.
Veletsos, A. S., and Nair, V. V. D., 1974, “Torsional
Vibration for Viscoelastic Foundations,” Journal of
Geotechnical Engineering, ASCE, V. 100, No. GT3, Mar.,
pp. 225-246.
Veletsos, A. S., and Verbic, B., 1973, “Vibration of
Viscoelastic Foundations,” Earthquake Engineering and
Structural Dynamics, V. 2, No. 1, July-Sept., pp. 87-102.
Veletsos, A. S., and Wei, Y. T., 1971, “Lateral and
Rocking Vibrations of Footings,” Journal of the Soil
Mechanics and Foundations Division, ASCE, V. 97, No. SM9,
Sept., pp. 1227-1248.
Wolf, J. P., and Darbre, G. R., 1984, “Dynamic-Stiffness
Matrix of Soil by the Boundary-Element Method: Embedded
Foundations,” Earthquake Engineering and Structural
Dynamics, V. 12, No. 3, May, pp. 401-416.
6.3—Software sources and other references
ANSYS—ANSYS, Inc., Global Headquarters
Southpointe
275 Technology Drive
Canonsburg, PA 15317
Web: http://www.ansys.com
DYNA—Dept. of Civil Engineering
Geotechnical Research Centre
London, Ontario, N6A 5B9
Canada
Web: http://www.engga.uwo.ca/civil/grc/computer.html
GTSTRUDL—Georgia Tech—CASE Center
School of Civil & Environmental Engineering
Atlanta, GA 30332-0355
Web: http://www.gtstrudl.gatech.edu/
RISA—RISA Technologies
26632 Towne Centre Drive, Suite 210
Foothill Ranch, CA 92610
Web: http://www.risatech.com/
SACS—Engineering Dynamics, Inc.
2113 38th Street
Kenner, LA 70065
Web: http://www.sacs-edi.com
FOUNDATIONS FOR DYNAMIC EQUIPMENT
SAP2000—Computers & Structures, Inc.
1995 University Ave., Suite 540
Berkeley, CA 94704
Web: http://www.csiberkeley.com
STAAD—Research Engineers International, Headquarters
22700 Savi Ranch Parkway
Yorba Linda, CA 92887-4608
Web: http://www.reiworld.com
Matlock, H.; Foo, H. C.; and Bryant, L. M., 1978, “Simulation of Lateral Pile Behavior Under Earthquake Motion,”
Earthquake Engineering and Soil Dynamics: Proceedings of
the ASCE Geotechnical Engineering Division Specialty
Conference, June 19-21, 1978, Pasadena, CA, ASCE.
Novak, M.; Nogami, T.; and Aboul-Ella, F., 1978, “Dynamic
Soil Reactions for Plain Strain Case,” Journal of Engineering
Mechanics, ASCE, V. 104, No. EM4, Aug., pp. 953-959.
6.4—Terminology
The following terms are common terminology for
dynamic equipment foundations. These terms may differ
from standard ACI terminology (ACI 116R).
acceleration—Time rate of change of velocity; a vector
quantity measured in units of gravity, in./s/s (m/s/s).
amplitude—Maximum value of an oscillating quantity
measured from the position or level of equilibrium (zero-topeak value). Amplitude may be expressed in terms of
displacement, velocity, acceleration, force, or any other
time-varying quantity. In general, the terms “single,” “zeroto-peak,” and “peak” are unnecessary modifiers for “amplitude.” However, the vibration measurement industry often
reports displacement measurements in terms of “peak-topeak” or “double” amplitude, although these are mathematical
misnomers. Thus, the “single” amplitude modifiers are used
to provide specificity.
analysis, dynamic—A general term referring to the process
of analyzing a vibratory system for evaluating its natural
frequencies, mode shapes, and responses to excitation.
analysis, forced response—That part of dynamic analysis
carried out to evaluate the response of a vibratory system
subjected to general excitation. The response of the system is
described by the complete integral of the governing differential
equation. Harmonic and time-history analyses are subsets of the
more general forced response analysis.
analysis, harmonic—The dynamic analysis of a vibratory
system subjected to sinusoidal-type excitation. Harmonic
analysis neglects the initial conditions and involves only the
particular solution to the governing differential equation
of motion.
analysis, modal—The dynamic analysis of a multidegreeof-freedom system by which the responses in each mode of
vibration are determined separately and then superimposed
to obtain the total response.
analysis, steady-state response—A term synonymous
with “harmonic analysis.”
analysis, time history—The dynamic analysis in which
the response of a vibratory system is evaluated based on a set
351.3R-61
of specified time-varying excitation parameters such as
force, acceleration, or displacement. The excitation input is
in the form of a time history for a specified duration of
interest. Time history analysis is also known as transient
response analysis.
center of gravity—That point within a body through
which, for any orientation of the body, passes the resultant of
the gravitational forces (weights); in common practice,
equivalent to center of mass or mass centroid; common notation: CG.
chock—Part of the interface between the machine and the
concrete foundation that provides final alignment capability
and adjustability.
critical speed—The speed (usually in rpm) of a rotating
element at which the element exhibits dynamic instability
and large amplitude motion. This motion develops when the
angular speed of the rotating element matches one of the
rotating element’s natural frequencies.
damping constant—The ratio between the damping force
and the system velocity. For viscously damped systems, the
ratio is constant (independent of frequency). Measured in units
of lbf-s/in. or (N-s/m); common notation: c.
damping ratio—The ratio of the actual system damping
to the system’s critical damping. A system’s critical
damping is the minimum amount of viscous damping that
leads to a system not oscillating in free vibration situations.
damping, geometric—The dissipation of energy
resulting from a reduction in intensity of mechanical waves
radiating from a vibration source and propagating through an
elastic medium, also called radiation damping.
damping, hysteretic—See material damping.
damping, material—The dissipation of energy within
material as a result of internal friction. The damping force is
directly proportional to the displacement, independent of
frequency, and in phase with the velocity of the system.
Material damping may also be called hysteretic damping,
structural damping, or internal damping. For consistency,
use of the term “material damping” is recommended.
damping, radiation—A variant term for geometric
damping.
damping, viscous—A type of damping in which the
damping force is directly proportional to the velocity of the
system at all times. Thus, the damping force is in phase with
the velocity.
degrees of freedom—The independent coordinates used
to specify the position of a body (or system) in motion at any
time. (See also single degree-of-freedom system and multidegree-of-freedom system.)
displacement—A vector quantity that specifies the position
of a body in motion with respect to a defined system of
reference, usually a position of rest or equilibrium. Rotational
(angular) displacement is measured in radians. Translational
displacement is measured in inches or meters. Peak-to-peak
displacement reflects the total travel that a point undergoes
through a full cycle of vibration. For simple harmonic
motion, this is twice the displacement amplitude. This quantity is commonly reported by vibration measuring equipment
and is often incorporated into acceptance criteria.
351.3R-62
ACI COMMITTEE REPORT
dynamic force—See excitation force; for discussion of
forces, also see the definition of excitation.
elastic half-space—A term used to describe an idealized
semi-finite medium such as soil mass.
excitation—Mechanical disturbance imparted to a physical
system by the direct application of an external force or
support motion.
excitation force—generated by the source of excitation.
These are frequency-dependent forces produced by dynamic
equipment. Force is usually express as pounds-force or
Newtons.
In English Units (Gravitational System): pound-force (lbf)
is the unit of force that, when acting on a mass of one slug,
will impart to it an acceleration of 1.0 ft/s/s. One kip (K) is
1000 lbf.
In SI Units (absolute System): newton (N) is the unit of
force that, when acting on a mass of one kilogram, will
impart to it an acceleration of 1.0 m/s/s. One kilo-Newton
(kN) is 1000 N.
In North America, pound is often taken to mean poundforce (lbf). It may, however, sometimes cause confusion
with pound-mass (lbm).
frequency—Number of complete vibrations per second.
The unit is cycles per second (cps) or Hertz (Hz). The latter
term is more commonly used; common notion: f.
frequency, circular—Frequency expressed in radians per
second (rad/s). Sometimes called angular frequency or
eigenvalue; common notation: ω.
frequency, excitation—The frequency of the vibration
source.
frequency, fundamental—The lowest natural frequency
of the vibratory system. This term may be used in a directiondependent manner so that the vertical fundamental
frequency may differ from a lateral fundamental frequency.
frequency, natural—The frequency at which a vibratory
system will vibrate without the influence of any external
force or damping. May be expressed in terms of frequency
(cps or Hz) or circular frequency (rad/s).
frequency ratio—The ratio of excitation frequency to the
natural frequency of the system.
frequency, resonant—The frequency at which the
dynamic magnification is a maximum.
geophone—A velocity transducer often used in soil tests.
inertia block—A solid, heavy piece, often made of
concrete and assumed to be rigid, that is used as part of the
foundation system principally for its weight contribution.
machine foundation—This term has always caused some
confusion as to exactly what it means. To the structural or
geotechnical engineer, foundation is the concrete substructure
resting on or below ground. The machine manufacturer,
however, has traditionally used this term to designate the
structural supporting system below the soleplate of the
machine. Thus, the elevated concrete structure supporting a
machine such as the paper machine is still called a machine
foundation by the manufacturer; although, such foundation
can be as high as 25 ft above ground. Such confusion may
lead to misinterpretation by the structural engineer of the
manufacturer’s design parameters and criteria. Within this
document the manufacturer’s understanding is used: the
supporting structure below the soleplate of the machine.
mass—The physical quality of a body reflecting the quantity of matter inherent in the body. A specific body’s mass
does not vary whereas the body’s weight will change
depending on the local gravitational field (on earth versus on
the moon). In vibration applications, a body’s mass is a critical parameter. Units associated with mass are typically
grams or kilograms in absolute SI units and pound-mass
(lbm) or slugs in FPS systems. A slug is a mass that, when
subjected to a force of one pound, accelerates one ft/s2. (1 slug
= 1 lbf-s2/ft). A pound-mass, when subjected to a force of
one pound, accelerates 32.2 ft/s2 (9.81 m/s2). That is, on
earth, a mass of one lbm weighs one lbf. Further explanation
can be found in ASTM SI-10.
mathematical model—The idealized representation of a
physical system for mathematical treatment and computer
analysis. The level of complexity should be compatible with
the required degree of accuracy. For dynamic analysis, a
mathematical model may include the following information:
• Geometry of the structure (joint coordinates);
• Types of members (such as beam, truss, plate, membranes, and solids), and physical properties (such as
section, sizes, and thickness);
• Concentrated masses;
• Member connectivity, end-fixity, and boundary conditions;
• Material properties (such as mass densities and elastic
constants);
• Damping; and
• Excitation forces, their locations, magnitudes, and
frequencies.
mode of vibration—A characteristic deflection shape of
a structure corresponding to a specific natural frequency of
the system, also known as an eigenvector.
modulus of elasticity—The ratio of axial stress to axial
strain, also known as Young’s Modulus.
modulus of subgrade reaction—The soil pressure per
unit displacement of a foundation (usually in the vertical
direction), also called coefficient of subgrade reaction. This
soil property is strain-rate dependent, therefore, different
values apply for static and dynamic loading.
modulus, shear—The ratio of shear stress to shear strain,
also know as modulus of rigidity; however, the term shear
modulus is more common and is always used in discussion
involving soils.
multi-degree-of-freedom system (MDOF)—A vibratory
system that requires two or more independent coordinates to
completely specify its motion. (See also single degree-offreedom system.)
resonance—The state of steady-state vibration in which
the excitation frequency is equal to or close to a system
damped natural frequency. In this state, system displacements
are usually amplified (for lightly damped systems) and can
be very sensitive to changes in the excitation frequency,
system stiffness, or system mass.
response—The motion of a vibratory system resulting from
excitation. The motion can be mathematically described in terms
of displacement, velocity, acceleration, or other parameters.
FOUNDATIONS FOR DYNAMIC EQUIPMENT
root-mean-square (rms) value—A time-weighted
average measurement of a particular quantity (force,
displacement, velocity, or acceleration). Rms velocity is a
useful measure of vibration severity in cases where the
vibration is complex such that displacement, velocity, and
acceleration are not clearly related. For general functions, a
rms value is calculated as
T
v rms =
1 2
--- v ( t ) dt
T
∫
0
For single-frequency, harmonic functions, an rms value is
equal to 0.707 of the function amplitude.
single degree-of-freedom system (SDOF)—A vibratory
system the motion of which can be completely specified by
351.3R-63
a single coordinate. (See also Multi-degree-of-freedom
system.)
soleplate—The member that interfaces to or between the
machine and the supporting structure.
stiffness—The ratio of the applied force to the resulting
foundation movement, expressed in lb/in. or (N/m); common
notation: k.
transmissibility—A measure of the ability of a dynamic
system to transmit energy from a source to a location,
commonly computed as a transmissibility ratio.
unbalanced force—See excitation force.
velocity—A vector quantity that identifies the time rate of
change of motion. Translational velocity is measured in in./s
(m/s), rotational velocity is rarely used.